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Polar optical phonon states and Fröhlich electron–phonon interaction Hamiltonians in a wurtzite nitride quantum dot
Physics Letters A 373 (2009) 2087–2090
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Polar optical phonon states and Fröhlich electron–phonon interaction Hamiltonians in a wurtzite nitride quantum dot L. Zhang 1 Department of Mechanism and Electron, Guangzhou Panyu Polytechnic, Guangzhou 511483, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 18 February 2009 Received in revised form 2 April 2009 Accepted 9 April 2009 Available online 18 April 2009 Communicated by A.R. Bishop PACS: 81.05.Ea 78.67.Hc 63.22.-m 63.20.Kd Keywords: Wurtzite nitride quantum dots Phonon states Electron–phonon interactions
a b s t r a c t Within the framework of the macroscopic dielectric continuum model, the surface-optical-propagating (SO-PR) mixing phonon modes of a quasi-zero-dimensional (Q0D) wurtzite cylindrical quantum dot (QD) structure are derived and studied. The analytical phonon states of SO-PR mixing modes are given. It is found that there are two types of SO-PR mixing phonon modes, i.e. ρ -SO/z-PR mixing modes and the z-SO/ρ -PR mixing modes existing in Q0D wurtzite QDs. And each SO-PR mixing modes also have symmetrical and antisymmetrical forms. Via the standard procedure of field quantization, the Fröhlich Hamiltonians of electron–(SO-PR) mixing phonons interaction are obtained. And the orthogonal relations of polarization eigenvectors for these SO-PR mixing modes are also displayed. Numerical calculations on a wurtzite GaN cylindrical QD are carried out. The results reveal that the dispersive frequencies of all the SO-PR mixing modes are the discrete functions of phonon wave-numbers and azimuthal quantum numbers. The behaviors that the SO-PR mixing phonon modes in wurtzite QDs reduce to the SO modes and PR modes in wurtzite quantum well (QW) and quantum well wire (QWR) systems are analyzed deeply from both of the viewpoints of physics and mathematics. The result shows that the present theories of polar mixing phonon modes in wurtzite cylindrical QDs are consistent with the phonon modes theories in wurtzite QWs and QWR systems. The analytical electron–phonon interaction Hamiltonians obtained here are useful for further analyzing phonon influence on optoelectronics properties of wurtzite Q0D QD structures. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Crystal lattice dynamical properties of nitride semiconductor low-dimensional quantum structures, such as quasi-2-dimensional (Q2D) quantum wells (QWs), quasi-1-dimensional (Q1D) quantum wires (QWRs) have attracted a lot of attentions both in theoretical and experimental investigations [1–7]. The driving force behind these efforts lies in the evident fact that lattice vibrations have important influence on the optoelectronics and electronics properties of nitride low-dimensional quantum systems [6,7]. In fact, 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 nitride materials. Hence lattice dynamical properties of nitride low-dimensional quantum structures, especial the Q2D QW and Q1D QWR structures have been intensively studied during the last two decades [1–7]. With the technological advancement of the crystal growing, not only Q2D nitride QW and Q1D nitride QWR structures, but also
1
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quasi-0-dimensional (Q0D) nitride quantum dots (QDs) can be fabricated [8–14]. It is well known that group-III nitride usually crystallize in the hexagonal wurtzite structure, whose physical properties behavior anisotropic in space. The previous works on the wurtzite QW and QWR structures reveal that the lattice vibrating modes (i.e. phonon modes) and electron–phonon interactions become more complicated with the increase of confined dimensionality [1–7]. Though the bound electronic states, excitonic states, donor bound excitons, as well as the nonlinear optical properties in the Q0D wurtzite QD have been widely investigated [11–14], the polar optical phonon states and their coupling features with electrons in wurtzite Q0D QD systems have rarely been reported due to the complexity of phonon modes in the structures originated from the high confined dimensionality and anisotropic wurtzite structure [15–17]. Thus we will investigate the polar phonon states and corresponding electron–phonon interaction Hamiltonians of a wurtzite Q0D QD in the present Letter. In fact, based on the dielectric continuum model (DCM), the optical phonon states and their dispersive properties as well as electron–phonon coupling functions in zinc-blende Q0D cylindrical QDs have been deduced and analyzed. To the best of our knowledge, apart from the recent works [15–17], there has little research on the polar phonon modes in wurtzite Q0D QD systems by now.
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With the aid of the DCM and Loudon’s uniaxial crystal model [18], we extended the works of polar optical phonon modes from the wurtzite Q2D QWs and Q1D QWRs to the wurtzite Q0D QDs structures [17]. But the dispersive properties of phonon modes in the wurtzite QDs have not been discussed, and electron–phonon coupling functions still have not been deduced. The present work is just to supply a gap of optical phonon mode theories in wurtzite QD structures. The main accomplishments and significance of this work are as follows. (i) The explicit phonon states and dispersive equations of one important optical phonon modes, i.e. the surface optical (SO)propagating (PR) phonon mixing modes are given. The difference and relationship of the phonon dispersive equations in wurtzite quantum QW, QWR and QD structures are analyzed in depth from both the viewpoints of physics and mathematics. (ii) It is found that the dispersion frequencies of the mixing phonon modes in wurtzite QDs are discrete functions of the azimuthal quantum number and axial wave-number of the cylindrical QD structures, which are obviously different from the cases of wurtzite Q2D QW and Q1D QWR structures. (iii) The orthogonal relation of polarization eigenvector of SOPR mixing phonon modes are obtained, and the Fröhlich electron–phonon interaction Hamiltonians are also deduced. The analytical results obtained are important and useful for further investigation of polaronic effect on the influence of electronic and optical properties in wurtzite Q0D QD structures. 2. Phonon states of SO-PR mixing modes and their dispersive equations of GaN cylindrical QDs It is well known that the wurtzite GaN QDs often have different shapes and symmetries, such as spherical-caps, spheroids, hexagonal pyramids and cylindroid structures [8–10]. For simplicity, we consider a freestanding wurtzite cylindrical QD structure with radius R and height 2d along the z-direction. The z-axis is taken to be along the direction of the c-axis of the wurtzite material and denote the radial- (axial-) direction as t (z). As the first step of solving the complicated mixing optical phonon modes in wurtzite cylindrical QDs, we will pay attention to the SO-PR mixing phonon modes only in the next text. The SO-PR mixing mode is a mode which behaves as SO mode in t ( z)-direction, and shows as PR mode in z(t )-direction. Considering the exchange of t-direction and z-direction, it is found that the SO-PR mixing modes also have two forms, i.e. the z-SO/ρ -PR and ρ -SO/ z-PR mixing modes. Under the cylindrical coordinates, the electrostatic potential functions of z-SO/ρ -PR mixing modes are given by SO-PR Φm (r) = eimϕ f PR (ρ )φ SO (z), a1 J m (kt1 ρ ), ρ R, PR f (ρ ) = a2 J m (kt2 ρ ) + a3 Y m (kt2 ρ ), ρ > R , ⎧ ⎧ b exp(k z), z < −d, z2 ⎨ 1 ⎪ ⎪ ⎪ ⎪ b2 sinh(k z1 z), | z| d, AS, ⎪ ⎪ ⎪ ⎨ ⎩ −b exp(−k z), z > d, 1 z2 φ SO (z) = ⎧ ⎪ z < −d, ⎪ ⎨ b1 exp(k z2 z), ⎪ ⎪ ⎪ ⎪ S. b cosh(k z1 z), | z| d, ⎪ ⎩⎩ 2 b1 exp(−k z2 z), z > d,
a2 K m (kt2 ρ ),
ρ R, ρ > R,
(2)
In Eq. (1), J m (x) [Y m (x)] is the Bessel [Neumann] function of morder. In Eq. (2), K m (x) [I m (x)] is the first- [second-] kind modified Bessel functions of order m. And ai and b i are coupling coefficients of phonon modes determining by additional boundary conditions (BCs). The symbols “AS” (“S”) in Eqs. (1) and (2) corresponds to the antisymmetrical (symmetrical) solution. This treatment completely satisfies the symmetry demand of the phonon potential in z-direction. According to the relationship of Loudon’s uniaxial crystal model [18] and the Laplace equation in the areas of the inner and outer QD, the dependent relations of the phonon wavenumbers kuv (u = t , z; v = 1, 2) can be chosen as
ti (ω)kti ± zi (ω)k zi = 0,
z1 (ω)k z1 ± z2 (ω)k z2 = 0.
(3)
In Eq. (3), t (ω) and z (ω) are the dielectric functions in tdirection and z-direction, respectively. The subscripts 1 and 2 in Eqs. (1), (2) and (3) correspond to the GaN material and vacuum dielectric environment, respectively. The reasonableness of Eq. (3) lies in the fact that the present theories in wurtzite Q0D QDs can reduce naturally to the corresponding results of Q1D QWR and Q2D QW structures well known under a certain condition, which will be discussed in detail later. By using the continuity BCs of the potential functions and the electric displacement vector at the axial interfaces z = ±d and the radial surface ρ = R, one can get the following two dispersive equations for the ρ -SO/ z-PR mixing modes [4,6], i.e. k z1 = nπ /2d,
n = ±1, ±2, . . . , t ,1kt1 K m (kt2 R ) I m−1 (kt1 R ) + I m+1 (kt1 R ) = −t2 kt2 I m (kt1 R ) K m−1 (kt2 R ) + K m+1 (kt2 R ) .
(4)
(5)
In Eq. (4), n taking even (odd) number corresponds to the symmetric (antisymmetric) ρ -SO/ z-PR mixing phonon modes. Connecting Eqs. (3)–(5), the dispersive frequencies of ρ -SO/ z-PR mixing modes can be worked out. In the same way, via the continuity BCs of the potential functions and the electric displacement vector at the interfaces, one can obtain another two dispersive equations (6) and (7) for the z-SO/ρ -PR mixing modes [3,7]. They are given by
√
arctan h −z2 /z1 /d, S,
√ kt2 t ,2 J m (kt1 R ) J m−1 (kt2 R ) − J m+1 (kt2 R ) + kt1 t ,1 J m (kt2 R ) J m+1 (kt1 R ) − J m−1 (kt1 R ) Y m (kt2 L )
= J m (kt2 L ) kt1 t ,1 J m+1 (kt1 R ) − J m−1 (kt1 R ) Y m (kt2 R ) + kt2 t ,2 Y m+1 (kt2 R ) − Y m−1 (kt2 R ) J m (kt1 R ) .
(1)
a1 I m (kt1 ρ ),
⎧ ⎧ b exp(ik z), z < −d, z2 ⎨ 1 ⎪ ⎪ ⎪ b sin(k z), ⎪ | z| d, AS, ⎪ 2 z1 ⎪ ⎪ ⎨ ⎩ −b exp(ik z), z > d, 1 z2 φ PR (z) = ⎧ ⎪ ⎪ ⎨ b1 exp(ik z2 z), z < −d, ⎪ ⎪ ⎪ ⎪ S. b cos(k z1 z), | z| d, ⎪ ⎩⎩ 3 b1 exp(ik z2 z), z > d,
k z1 =
For the ρ -SO/ z-PR mixing modes, their electrostatic potentials can be written as SO-PR Φm (r) = e imϕ f SO (ρ )φ PR (z),
f SO (ρ ) =
arctan h −z1 /z2 /d, AS,
(6)
(7)
In Eq. (7), L is the maximum radial size of the nonpolar dielectric environment (in general, L R). 3. Free phonon fields and Fröhlich electron–phonon interaction Hamiltonians To obtain the expressions for the Hamiltonian of the free phonon field and electron–phonon interaction Hamiltonian, we
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first institute the orthonormality relationships of polarization vector of SO-PR mixing phonon modes. Using the formula P = (1 − )/4π ∇Φ(r), and via Eq. (1), we obtain the orthogonal relations of polarization vector for the antisymmetrical z-SO/ρ -PR mixing modes, i.e.
∗
-PR SO-PR 3 PSO AS,m · PAS,m d r
=
| A 0 |2
2 ρ dρ dz b22 sinh(k z1 z)2 (1 − t1 )2kt1
16π
2 2 × Jm −1 (kt1 ρ ) + J m+1 (kt1 ρ )
2 (kt1 ρ )(1 − z1 )2 k2z1 b22 cosh(k z1 z)2 δm m . + 2 Jm
(8)
If exchanging the position of sinh(k z1 z) and cosh(k z1 z) in Eq. (8), the orthogonal relations of polarization vector for the symmetrical z-SO/ρ -PR mixing modes is obtained. Based on Eq. (2), the orthogonal relations of polarization vectors for symmetrical and antisymmetrical ρ -SO/ z-PR mixing modes are unified as
Fig. 1. Dispersive frequencies ω of the ρ -SO/ z-PR mixing modes as a function of the wave-number k z1 for three azimuthal quantum numbers m = 0, 1, 2.
∗
-PR SO-PR 3 PSO AS/S,m · PAS/S,m d r
=
| A 0 |2 16π
2 ρ dρ dz b2 cos(k z1 z) + b3 sin(k z1 z) (1 − t1 )2
4. Numerical results and discussion
2 2 2 2 × kt1 Km (kt2 R ) I m −1 (kt1 ρ ) + I m+1 (kt1 ρ ) 2 2 (kt2 R ) I m (kt1 + 2K m
2
ρ )(1 − z1 )
2 × k2z1 b2 sin(k z1 z) + b3 cos(k z1 z)
δm m .
(9)
When deducing the above orthogonal relations of polarization vectors (8), (9), only the region inside the cylindrical QDs are considered, and the region outside the QDs are neglected because the polarization vectors in this region is null due to d = 1. Furthermore, it is also observed from these equations that only the azimuthal quantum number m is good quantum number originating from the symmetry of cylindrical QDs, which is obviously different from the situations of QWs and QWRs systems [1–3,6,7,16]. Following the quantization steps similar to those in Refs. [1–4], one can obtain the free phonon Hamiltonian operators for the SO-PR mixing phonons, i.e. H SO-PR =
†
h¯ ω bm (k z1 )bm (k z1 ) +
m,k z1
1 2
,
(10)
†
where bm (k z1 ) and bm (k z1 ) are creation and annihilation operators for the SO-PR mixing phonons of mth modes. The interaction Hamiltonians of electron with the SO-PR phonon fields is read as H e-ph = −e Φ SO-PR (r). Via expanding Φ SO-PR (r) in terms of the normal modes, and after some trivial algorithms, we get the electron– (SO-PR)-phonons interaction Hamiltonians as H e-ph = −
ΓmSO,k-z1PR (ρ )ΓmSO,k-z1PR (z) bm (k z1 )eimϕ + H.c. ,
(11)
m,k z1
-PR SO-PR where ΓmSO ,k z1 (ρ ) and Γm,k z1 ( z) are the coupling functions defined as
ΓmSO,k-z1PR (ρ ) = Nm (k z1 ) f SO,PR (ρ ),
ΓmSO,k-z1PR (z) = Nm (k z1 ) φ SO,PR (z), with
Nm (k z1 ) =
(12)
h¯ e 2
ω
| A 0 |.
(13)
In Eq. (12), the functions f SO,PR (ρ ) and φ SO,PR ( z) are defined in Eq. (1) and Eq. (2).
The height 2d and the radius R of the cylindrical QD structure are chosen as d = a B and R = 2a B , respectively (a B is the effective Bohr radius of the wurtzite GaN bulk material, which is approximately 2.4 nm). The physical parameters of the materials used in our calculation are the same as in Refs. [2,3]. In Fig. 1, the dispersive frequencies ω of the ρ -SO/ z-PR mixing modes as a function of the wave-number k z1 in z-direction are plotted for m = 0, 1, 2. From the figure, it can be seen that the dispersive frequencies of the SO-PR mixing phonon modes are the discrete functions of the wave-number k z1 , which is quite similar to the energy levels of electronic states in Q0D QD structures. This is the typical feature of confined phonon modes in Q0D QD systems [9], and it is obviously different from the dispersive characteristics of confined phonon modes in wurtzite Q2D QWs [1–4] and Q1D QWRs structures [6,7]. In wurtzite Q2D QWs (Q1D QWRs) systems, the dispersive frequencies of confined phonon modes are always continuous functions of the plane (axial) wave-numbers due to their translational symmetry [1–4,6,7]. But for the wurtzite cylindrical QDs, they have no translational symmetry. Thus the dispersive frequencies of phonon modes in wurtzite cylindrical QDs just take a series of discrete values. We also notice that all the intervals of adjacent kt1 for the ρ -SO/ z-PR mixing phonon modes are equal, which can be fully understood from the dispersion Eq. (4). For a certain phonon branch of ρ -SO/ z-PR modes (m is unchanged), the dispersive frequencies are the monotonic and incremental functions of k z1 . Moreover, with the increase of m, the dispersions of ρ -SO/ z-PR mixing phonon modes become weaker and weaker. As kt1 approaches ∞, the value 715.42 cm−1 is the limiting frequency of the ρ -SO/ z-PR mixing phonon modes. In fact, based on the limiting relations of I m (x) and K m (x) [19], it is easy to prove that Eq. (5) will degenerate to the form of t1 = 1 as kt1 → ∞. This equation just gives the limited frequency of 715.42 cm−1 . These are the general features of phonon modes in lowdimensional quantum structures [1–4,6,7]. From Eq. (4), it is found that the wave-number k z1 will become continuous as d → ∞. Under this condition, based √ on Eq. (3) one can get the relations: k z1 = k z2 = k z and kti = zi (ω)/ti (ω)k z . Substituting the two conditions into Eq. (5), Eq. (5) will reduce to the form of dispersive equation which is completely same as the dispersive equation of SO phonon modes in Q1D wurtzite QWRs [6]. This explains distinctly the fact, from a mathematic view of point, that the SO-PR mixing phonon modes in Q0D wurtzite
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tain limited frequency for very large wave-number. These are the typical characteristics of SO modes in QW structures [1,3]. In addition, for a fixing wave-number or azimuthal quantum number, there exists infinite SO-PR phonon modes in wurtzite QDs. This is just the classical feature of the PR modes in wurtzite QWRs [7]. Hence the SO-PR mixing modes in wurtzite QDs exhibits both the behaviors of SO modes and PR modes in quantum systems, which is also a reason named these modes as SO-PR mixing modes. In fact, when d approaches ∞, it will result in the continual wavenumber k z1 via Eq. (6). Thus the Eq. (7) will reduce to the dispersive equation of the PR modes in Q1D wurtzite QWRs [7]. This is the natural result because the cylindrical QDs will degenerate into cylindrical QWRs when the height of the QDs approach infinity. 5. Conclusions
Fig. 2. Dispersive frequencies ω of symmetrical (nether curve) and antisymmetrical (upper curve) z-SO/ρ -PR mixing phonon modes in wurtzite GaN QD systems as a function of k z1 .
QDs reduce to the SO phonon modes in Q1D wurtzite QWRs under the condition of d → ∞. From a pure physical viewpoint, as the height d approaches ∞ and the radius R is kept at a finite value, the physical model of the wurtzite Q0D cylindrical QD structures will naturally reduce to the wurtzite Q1D cylindrical QWR structures. This further proves the correctness and reliability of the phonon modes theories in Q0D wurtzite QD systems established in the present work. The dispersive curves of z-SO/ρ -PR mixing phonon modes as a function of k z1 are depicted in Fig. 2. Three different azimuthal quantum number m = 0, 1, 2 are chosen. It is observed clearly that there are two branches of z-SO/ρ -PR mixing modes in GaN cylindrical QDs. The high frequency branch (the upper branch) is the antisymmetrical z-SO/ρ -PR mixing phonon modes, and the low frequency branch (the nether branch) is the symmetrical z-SO/ρ -PR mixing modes. Same as the situation of ρ -SO/z-PR mixing phonon modes, the dispersive frequencies of z-SO/ρ -PR mixing modes are also the discrete functions of k z1 . The dispersive frequencies of the high (low) frequency z-SO/ρ -PR mixing modes are the monotonic and degressive (incremental) functions of k z1 . As k z1 approaches infinity, the two branches of z-SO/ρ -PR mixing modes converge to the frequency of 706.03 cm−1 . This limiting value is determined by the equation z1 − 1 = 0. As k z1 → ∞, Eq. (7) can also reduce to the form of z1 − 1 = 0. Compared the symmetrical z-SO/ρ -PR mixing modes with the antisymmetrical z-SO/ρ -PR mixing modes, it is found that the dispersions of the symmetrical modes are more obvious than those of the antisymmetrical modes. On the other hand, as m is fixing, the intervals of adjacent k z1 are unequal. The intervals decrease with the increase of k z1 , which differ obviously from the cases of ρ -SO/ z-PR mixing modes (refer to Fig. 1). Moreover, the larger the azimuthal quantum number m, the adjacent intervals of k z1 become smaller. Comparing the dispersive features of z-SO/ρ -PR mixing modes in GaN QD with those of the phonon modes in GaN QWs [1–4] and QWRs systems [6,7], we find that the z-SO/ρ -PR mixing modes in wurtzite cylindrical QDs possess both characteristics of the SO phonon modes in wurtzite QWs and the PR phonon modes in wurtzite QWRs structures. For example, the SO-PR mixing modes in wurtzite cylindrical QD systems have two branches. One is symmetrical, the other is antisymmetrical, and they converge to a cer-
In the present work, we have investigated the SO-PR mixing phonon modes in a wurtzite cylindrical QD system. It is found that there exist two types of SO-PR mixing phonon modes in wurtzite cylindrical QD structures, namely the ρ -SO/ z-PR mixing modes and the z-SO/ρ -PR mixing modes. And each SO-PR mixing phonon modes also have two forms. One is symmetrical, and the other is antisymmetrical. Based on the method of field quantization, the Fröhlich Hamiltonian of electron–(SO-PR) mixing phonons interactions are given. Numerical calculations on the dispersive properties of the SO-PR mixing phonon modes of a wurtzite GaN cylindrical QD are performed. We observed that the dispersive spectra of SO/PR mixing modes just only take a serial of discrete values due to the three-dimension confinement of the QD structure. As the height or the radius of the Q0D wurtzite cylindrical QDs approach infinity, both types of the SO-PR mixing modes will reduce to the SO modes or PR modes in Q2D wurtzite QW and Q1D QWR structures. These reducing behaviors of the mixing modes have been analyzed in depth from both of the viewpoints of physics and mathematics. This shows the theories of mixing phonon modes in wurtzite Q0D QDs established in the present Letter are consistent with those in wurtzite Q1D QW and Q0D QWR systems [1–4,6,7]. Therefore the present theories and numerical results are correct and reliable. 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 Q0D wurtzite QD systems. References [1] S.M. Komirenko, K.W. Kim, M.A. Stroscio, M. Dutta, Phys. Rev. B 59 (1999) 5013. [2] J. Gleize, M.A. Renucci, J. Frandon, F. Demangeot, Phys. Rev. B 60 (1999) 15985; J. Gleize, M.A. Renucci, J. Frandon, F. Demangeot, Mater. Sci. Eng. B 82 (2001) 27. [3] J.J. Shi, Phys. Rev. B 68 (2003) 165335. [4] L. Zhang, J.J. Shi, Commun. Theor. Phys. 45 (2006) 935. [5] L.T. Tan, R.W. Martin, K.P. O’Donnell, I.M. Watson, Appl. Phys. Lett. 89 (2006) 101910. [6] L. Zhang, J.J. Shi, T.L. Tansley, Phys. Rev. B 71 (2005) 245324. [7] L. Zhang, J.S. Liao, Superlatt. Microstruct. 43 (2008) 28. [8] A.F. Jarjoura, et al., Superlatt. Microstruct. 43 (2008) 431. [9] N. Garro, et al., Phys. Rev. B 74 (2006) 075305. [10] T. Xu, et al., J. Appl. Phys. 102 (2007) 073517. [11] J.J. Shi, T.L. Tansley, Solid State Commun. 138 (2006) 26. [12] L. Liu, J. Li, G. Xiong, Physica E 25 (2005) 466. [13] C.X. Wang, G.G. Xiong, Microelectron. J. 37 (2006) 847. [14] C.X. Xia, F. Jiang, S.Y. Wei, Current Appl. Phys. 8 (2008) 153. [15] V.A. Fonoberov, A.A. Balandin, Phys. Rev. B 70 (2004) 233205. [16] P.M. Chassaing, et al., Phys. Rev. B 77 (2008) 153306. [17] L. Zhang, J.J. Shi, H.J. Xie, Solid State Commun. 140 (2006) 549. [18] R. Loudon, Adv. Phys. 13 (1964) 423. [19] J.Y. Zeng, Quantum Mechanics, second ed., Science Press, Beijing, 2000, pp. 744–750 (in Chinese).
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Polar optical phonon states and Fröhlich electron–phonon interaction Hamiltonians in a wurtzite nitride quantum dot L. Zhang 1 Department of Mechanism and Electron, Guangzhou Panyu Polytechnic, Guangzhou 511483, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 18 February 2009 Received in revised form 2 April 2009 Accepted 9 April 2009 Available online 18 April 2009 Communicated by A.R. Bishop PACS: 81.05.Ea 78.67.Hc 63.22.-m 63.20.Kd Keywords: Wurtzite nitride quantum dots Phonon states Electron–phonon interactions
a b s t r a c t Within the framework of the macroscopic dielectric continuum model, the surface-optical-propagating (SO-PR) mixing phonon modes of a quasi-zero-dimensional (Q0D) wurtzite cylindrical quantum dot (QD) structure are derived and studied. The analytical phonon states of SO-PR mixing modes are given. It is found that there are two types of SO-PR mixing phonon modes, i.e. ρ -SO/z-PR mixing modes and the z-SO/ρ -PR mixing modes existing in Q0D wurtzite QDs. And each SO-PR mixing modes also have symmetrical and antisymmetrical forms. Via the standard procedure of field quantization, the Fröhlich Hamiltonians of electron–(SO-PR) mixing phonons interaction are obtained. And the orthogonal relations of polarization eigenvectors for these SO-PR mixing modes are also displayed. Numerical calculations on a wurtzite GaN cylindrical QD are carried out. The results reveal that the dispersive frequencies of all the SO-PR mixing modes are the discrete functions of phonon wave-numbers and azimuthal quantum numbers. The behaviors that the SO-PR mixing phonon modes in wurtzite QDs reduce to the SO modes and PR modes in wurtzite quantum well (QW) and quantum well wire (QWR) systems are analyzed deeply from both of the viewpoints of physics and mathematics. The result shows that the present theories of polar mixing phonon modes in wurtzite cylindrical QDs are consistent with the phonon modes theories in wurtzite QWs and QWR systems. The analytical electron–phonon interaction Hamiltonians obtained here are useful for further analyzing phonon influence on optoelectronics properties of wurtzite Q0D QD structures. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Crystal lattice dynamical properties of nitride semiconductor low-dimensional quantum structures, such as quasi-2-dimensional (Q2D) quantum wells (QWs), quasi-1-dimensional (Q1D) quantum wires (QWRs) have attracted a lot of attentions both in theoretical and experimental investigations [1–7]. The driving force behind these efforts lies in the evident fact that lattice vibrations have important influence on the optoelectronics and electronics properties of nitride low-dimensional quantum systems [6,7]. In fact, 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 nitride materials. Hence lattice dynamical properties of nitride low-dimensional quantum structures, especial the Q2D QW and Q1D QWR structures have been intensively studied during the last two decades [1–7]. With the technological advancement of the crystal growing, not only Q2D nitride QW and Q1D nitride QWR structures, but also
1
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quasi-0-dimensional (Q0D) nitride quantum dots (QDs) can be fabricated [8–14]. It is well known that group-III nitride usually crystallize in the hexagonal wurtzite structure, whose physical properties behavior anisotropic in space. The previous works on the wurtzite QW and QWR structures reveal that the lattice vibrating modes (i.e. phonon modes) and electron–phonon interactions become more complicated with the increase of confined dimensionality [1–7]. Though the bound electronic states, excitonic states, donor bound excitons, as well as the nonlinear optical properties in the Q0D wurtzite QD have been widely investigated [11–14], the polar optical phonon states and their coupling features with electrons in wurtzite Q0D QD systems have rarely been reported due to the complexity of phonon modes in the structures originated from the high confined dimensionality and anisotropic wurtzite structure [15–17]. Thus we will investigate the polar phonon states and corresponding electron–phonon interaction Hamiltonians of a wurtzite Q0D QD in the present Letter. In fact, based on the dielectric continuum model (DCM), the optical phonon states and their dispersive properties as well as electron–phonon coupling functions in zinc-blende Q0D cylindrical QDs have been deduced and analyzed. To the best of our knowledge, apart from the recent works [15–17], there has little research on the polar phonon modes in wurtzite Q0D QD systems by now.
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With the aid of the DCM and Loudon’s uniaxial crystal model [18], we extended the works of polar optical phonon modes from the wurtzite Q2D QWs and Q1D QWRs to the wurtzite Q0D QDs structures [17]. But the dispersive properties of phonon modes in the wurtzite QDs have not been discussed, and electron–phonon coupling functions still have not been deduced. The present work is just to supply a gap of optical phonon mode theories in wurtzite QD structures. The main accomplishments and significance of this work are as follows. (i) The explicit phonon states and dispersive equations of one important optical phonon modes, i.e. the surface optical (SO)propagating (PR) phonon mixing modes are given. The difference and relationship of the phonon dispersive equations in wurtzite quantum QW, QWR and QD structures are analyzed in depth from both the viewpoints of physics and mathematics. (ii) It is found that the dispersion frequencies of the mixing phonon modes in wurtzite QDs are discrete functions of the azimuthal quantum number and axial wave-number of the cylindrical QD structures, which are obviously different from the cases of wurtzite Q2D QW and Q1D QWR structures. (iii) The orthogonal relation of polarization eigenvector of SOPR mixing phonon modes are obtained, and the Fröhlich electron–phonon interaction Hamiltonians are also deduced. The analytical results obtained are important and useful for further investigation of polaronic effect on the influence of electronic and optical properties in wurtzite Q0D QD structures. 2. Phonon states of SO-PR mixing modes and their dispersive equations of GaN cylindrical QDs It is well known that the wurtzite GaN QDs often have different shapes and symmetries, such as spherical-caps, spheroids, hexagonal pyramids and cylindroid structures [8–10]. For simplicity, we consider a freestanding wurtzite cylindrical QD structure with radius R and height 2d along the z-direction. The z-axis is taken to be along the direction of the c-axis of the wurtzite material and denote the radial- (axial-) direction as t (z). As the first step of solving the complicated mixing optical phonon modes in wurtzite cylindrical QDs, we will pay attention to the SO-PR mixing phonon modes only in the next text. The SO-PR mixing mode is a mode which behaves as SO mode in t ( z)-direction, and shows as PR mode in z(t )-direction. Considering the exchange of t-direction and z-direction, it is found that the SO-PR mixing modes also have two forms, i.e. the z-SO/ρ -PR and ρ -SO/ z-PR mixing modes. Under the cylindrical coordinates, the electrostatic potential functions of z-SO/ρ -PR mixing modes are given by SO-PR Φm (r) = eimϕ f PR (ρ )φ SO (z), a1 J m (kt1 ρ ), ρ R, PR f (ρ ) = a2 J m (kt2 ρ ) + a3 Y m (kt2 ρ ), ρ > R , ⎧ ⎧ b exp(k z), z < −d, z2 ⎨ 1 ⎪ ⎪ ⎪ ⎪ b2 sinh(k z1 z), | z| d, AS, ⎪ ⎪ ⎪ ⎨ ⎩ −b exp(−k z), z > d, 1 z2 φ SO (z) = ⎧ ⎪ z < −d, ⎪ ⎨ b1 exp(k z2 z), ⎪ ⎪ ⎪ ⎪ S. b cosh(k z1 z), | z| d, ⎪ ⎩⎩ 2 b1 exp(−k z2 z), z > d,
a2 K m (kt2 ρ ),
ρ R, ρ > R,
(2)
In Eq. (1), J m (x) [Y m (x)] is the Bessel [Neumann] function of morder. In Eq. (2), K m (x) [I m (x)] is the first- [second-] kind modified Bessel functions of order m. And ai and b i are coupling coefficients of phonon modes determining by additional boundary conditions (BCs). The symbols “AS” (“S”) in Eqs. (1) and (2) corresponds to the antisymmetrical (symmetrical) solution. This treatment completely satisfies the symmetry demand of the phonon potential in z-direction. According to the relationship of Loudon’s uniaxial crystal model [18] and the Laplace equation in the areas of the inner and outer QD, the dependent relations of the phonon wavenumbers kuv (u = t , z; v = 1, 2) can be chosen as
ti (ω)kti ± zi (ω)k zi = 0,
z1 (ω)k z1 ± z2 (ω)k z2 = 0.
(3)
In Eq. (3), t (ω) and z (ω) are the dielectric functions in tdirection and z-direction, respectively. The subscripts 1 and 2 in Eqs. (1), (2) and (3) correspond to the GaN material and vacuum dielectric environment, respectively. The reasonableness of Eq. (3) lies in the fact that the present theories in wurtzite Q0D QDs can reduce naturally to the corresponding results of Q1D QWR and Q2D QW structures well known under a certain condition, which will be discussed in detail later. By using the continuity BCs of the potential functions and the electric displacement vector at the axial interfaces z = ±d and the radial surface ρ = R, one can get the following two dispersive equations for the ρ -SO/ z-PR mixing modes [4,6], i.e. k z1 = nπ /2d,
n = ±1, ±2, . . . , t ,1kt1 K m (kt2 R ) I m−1 (kt1 R ) + I m+1 (kt1 R ) = −t2 kt2 I m (kt1 R ) K m−1 (kt2 R ) + K m+1 (kt2 R ) .
(4)
(5)
In Eq. (4), n taking even (odd) number corresponds to the symmetric (antisymmetric) ρ -SO/ z-PR mixing phonon modes. Connecting Eqs. (3)–(5), the dispersive frequencies of ρ -SO/ z-PR mixing modes can be worked out. In the same way, via the continuity BCs of the potential functions and the electric displacement vector at the interfaces, one can obtain another two dispersive equations (6) and (7) for the z-SO/ρ -PR mixing modes [3,7]. They are given by
√
arctan h −z2 /z1 /d, S,
√ kt2 t ,2 J m (kt1 R ) J m−1 (kt2 R ) − J m+1 (kt2 R ) + kt1 t ,1 J m (kt2 R ) J m+1 (kt1 R ) − J m−1 (kt1 R ) Y m (kt2 L )
= J m (kt2 L ) kt1 t ,1 J m+1 (kt1 R ) − J m−1 (kt1 R ) Y m (kt2 R ) + kt2 t ,2 Y m+1 (kt2 R ) − Y m−1 (kt2 R ) J m (kt1 R ) .
(1)
a1 I m (kt1 ρ ),
⎧ ⎧ b exp(ik z), z < −d, z2 ⎨ 1 ⎪ ⎪ ⎪ b sin(k z), ⎪ | z| d, AS, ⎪ 2 z1 ⎪ ⎪ ⎨ ⎩ −b exp(ik z), z > d, 1 z2 φ PR (z) = ⎧ ⎪ ⎪ ⎨ b1 exp(ik z2 z), z < −d, ⎪ ⎪ ⎪ ⎪ S. b cos(k z1 z), | z| d, ⎪ ⎩⎩ 3 b1 exp(ik z2 z), z > d,
k z1 =
For the ρ -SO/ z-PR mixing modes, their electrostatic potentials can be written as SO-PR Φm (r) = e imϕ f SO (ρ )φ PR (z),
f SO (ρ ) =
arctan h −z1 /z2 /d, AS,
(6)
(7)
In Eq. (7), L is the maximum radial size of the nonpolar dielectric environment (in general, L R). 3. Free phonon fields and Fröhlich electron–phonon interaction Hamiltonians To obtain the expressions for the Hamiltonian of the free phonon field and electron–phonon interaction Hamiltonian, we
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first institute the orthonormality relationships of polarization vector of SO-PR mixing phonon modes. Using the formula P = (1 − )/4π ∇Φ(r), and via Eq. (1), we obtain the orthogonal relations of polarization vector for the antisymmetrical z-SO/ρ -PR mixing modes, i.e.
∗
-PR SO-PR 3 PSO AS,m · PAS,m d r
=
| A 0 |2
2 ρ dρ dz b22 sinh(k z1 z)2 (1 − t1 )2kt1
16π
2 2 × Jm −1 (kt1 ρ ) + J m+1 (kt1 ρ )
2 (kt1 ρ )(1 − z1 )2 k2z1 b22 cosh(k z1 z)2 δm m . + 2 Jm
(8)
If exchanging the position of sinh(k z1 z) and cosh(k z1 z) in Eq. (8), the orthogonal relations of polarization vector for the symmetrical z-SO/ρ -PR mixing modes is obtained. Based on Eq. (2), the orthogonal relations of polarization vectors for symmetrical and antisymmetrical ρ -SO/ z-PR mixing modes are unified as
Fig. 1. Dispersive frequencies ω of the ρ -SO/ z-PR mixing modes as a function of the wave-number k z1 for three azimuthal quantum numbers m = 0, 1, 2.
∗
-PR SO-PR 3 PSO AS/S,m · PAS/S,m d r
=
| A 0 |2 16π
2 ρ dρ dz b2 cos(k z1 z) + b3 sin(k z1 z) (1 − t1 )2
4. Numerical results and discussion
2 2 2 2 × kt1 Km (kt2 R ) I m −1 (kt1 ρ ) + I m+1 (kt1 ρ ) 2 2 (kt2 R ) I m (kt1 + 2K m
2
ρ )(1 − z1 )
2 × k2z1 b2 sin(k z1 z) + b3 cos(k z1 z)
δm m .
(9)
When deducing the above orthogonal relations of polarization vectors (8), (9), only the region inside the cylindrical QDs are considered, and the region outside the QDs are neglected because the polarization vectors in this region is null due to d = 1. Furthermore, it is also observed from these equations that only the azimuthal quantum number m is good quantum number originating from the symmetry of cylindrical QDs, which is obviously different from the situations of QWs and QWRs systems [1–3,6,7,16]. Following the quantization steps similar to those in Refs. [1–4], one can obtain the free phonon Hamiltonian operators for the SO-PR mixing phonons, i.e. H SO-PR =
†
h¯ ω bm (k z1 )bm (k z1 ) +
m,k z1
1 2
,
(10)
†
where bm (k z1 ) and bm (k z1 ) are creation and annihilation operators for the SO-PR mixing phonons of mth modes. The interaction Hamiltonians of electron with the SO-PR phonon fields is read as H e-ph = −e Φ SO-PR (r). Via expanding Φ SO-PR (r) in terms of the normal modes, and after some trivial algorithms, we get the electron– (SO-PR)-phonons interaction Hamiltonians as H e-ph = −
ΓmSO,k-z1PR (ρ )ΓmSO,k-z1PR (z) bm (k z1 )eimϕ + H.c. ,
(11)
m,k z1
-PR SO-PR where ΓmSO ,k z1 (ρ ) and Γm,k z1 ( z) are the coupling functions defined as
ΓmSO,k-z1PR (ρ ) = Nm (k z1 ) f SO,PR (ρ ),
ΓmSO,k-z1PR (z) = Nm (k z1 ) φ SO,PR (z), with
Nm (k z1 ) =
(12)
h¯ e 2
ω
| A 0 |.
(13)
In Eq. (12), the functions f SO,PR (ρ ) and φ SO,PR ( z) are defined in Eq. (1) and Eq. (2).
The height 2d and the radius R of the cylindrical QD structure are chosen as d = a B and R = 2a B , respectively (a B is the effective Bohr radius of the wurtzite GaN bulk material, which is approximately 2.4 nm). The physical parameters of the materials used in our calculation are the same as in Refs. [2,3]. In Fig. 1, the dispersive frequencies ω of the ρ -SO/ z-PR mixing modes as a function of the wave-number k z1 in z-direction are plotted for m = 0, 1, 2. From the figure, it can be seen that the dispersive frequencies of the SO-PR mixing phonon modes are the discrete functions of the wave-number k z1 , which is quite similar to the energy levels of electronic states in Q0D QD structures. This is the typical feature of confined phonon modes in Q0D QD systems [9], and it is obviously different from the dispersive characteristics of confined phonon modes in wurtzite Q2D QWs [1–4] and Q1D QWRs structures [6,7]. In wurtzite Q2D QWs (Q1D QWRs) systems, the dispersive frequencies of confined phonon modes are always continuous functions of the plane (axial) wave-numbers due to their translational symmetry [1–4,6,7]. But for the wurtzite cylindrical QDs, they have no translational symmetry. Thus the dispersive frequencies of phonon modes in wurtzite cylindrical QDs just take a series of discrete values. We also notice that all the intervals of adjacent kt1 for the ρ -SO/ z-PR mixing phonon modes are equal, which can be fully understood from the dispersion Eq. (4). For a certain phonon branch of ρ -SO/ z-PR modes (m is unchanged), the dispersive frequencies are the monotonic and incremental functions of k z1 . Moreover, with the increase of m, the dispersions of ρ -SO/ z-PR mixing phonon modes become weaker and weaker. As kt1 approaches ∞, the value 715.42 cm−1 is the limiting frequency of the ρ -SO/ z-PR mixing phonon modes. In fact, based on the limiting relations of I m (x) and K m (x) [19], it is easy to prove that Eq. (5) will degenerate to the form of t1 = 1 as kt1 → ∞. This equation just gives the limited frequency of 715.42 cm−1 . These are the general features of phonon modes in lowdimensional quantum structures [1–4,6,7]. From Eq. (4), it is found that the wave-number k z1 will become continuous as d → ∞. Under this condition, based √ on Eq. (3) one can get the relations: k z1 = k z2 = k z and kti = zi (ω)/ti (ω)k z . Substituting the two conditions into Eq. (5), Eq. (5) will reduce to the form of dispersive equation which is completely same as the dispersive equation of SO phonon modes in Q1D wurtzite QWRs [6]. This explains distinctly the fact, from a mathematic view of point, that the SO-PR mixing phonon modes in Q0D wurtzite
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tain limited frequency for very large wave-number. These are the typical characteristics of SO modes in QW structures [1,3]. In addition, for a fixing wave-number or azimuthal quantum number, there exists infinite SO-PR phonon modes in wurtzite QDs. This is just the classical feature of the PR modes in wurtzite QWRs [7]. Hence the SO-PR mixing modes in wurtzite QDs exhibits both the behaviors of SO modes and PR modes in quantum systems, which is also a reason named these modes as SO-PR mixing modes. In fact, when d approaches ∞, it will result in the continual wavenumber k z1 via Eq. (6). Thus the Eq. (7) will reduce to the dispersive equation of the PR modes in Q1D wurtzite QWRs [7]. This is the natural result because the cylindrical QDs will degenerate into cylindrical QWRs when the height of the QDs approach infinity. 5. Conclusions
Fig. 2. Dispersive frequencies ω of symmetrical (nether curve) and antisymmetrical (upper curve) z-SO/ρ -PR mixing phonon modes in wurtzite GaN QD systems as a function of k z1 .
QDs reduce to the SO phonon modes in Q1D wurtzite QWRs under the condition of d → ∞. From a pure physical viewpoint, as the height d approaches ∞ and the radius R is kept at a finite value, the physical model of the wurtzite Q0D cylindrical QD structures will naturally reduce to the wurtzite Q1D cylindrical QWR structures. This further proves the correctness and reliability of the phonon modes theories in Q0D wurtzite QD systems established in the present work. The dispersive curves of z-SO/ρ -PR mixing phonon modes as a function of k z1 are depicted in Fig. 2. Three different azimuthal quantum number m = 0, 1, 2 are chosen. It is observed clearly that there are two branches of z-SO/ρ -PR mixing modes in GaN cylindrical QDs. The high frequency branch (the upper branch) is the antisymmetrical z-SO/ρ -PR mixing phonon modes, and the low frequency branch (the nether branch) is the symmetrical z-SO/ρ -PR mixing modes. Same as the situation of ρ -SO/z-PR mixing phonon modes, the dispersive frequencies of z-SO/ρ -PR mixing modes are also the discrete functions of k z1 . The dispersive frequencies of the high (low) frequency z-SO/ρ -PR mixing modes are the monotonic and degressive (incremental) functions of k z1 . As k z1 approaches infinity, the two branches of z-SO/ρ -PR mixing modes converge to the frequency of 706.03 cm−1 . This limiting value is determined by the equation z1 − 1 = 0. As k z1 → ∞, Eq. (7) can also reduce to the form of z1 − 1 = 0. Compared the symmetrical z-SO/ρ -PR mixing modes with the antisymmetrical z-SO/ρ -PR mixing modes, it is found that the dispersions of the symmetrical modes are more obvious than those of the antisymmetrical modes. On the other hand, as m is fixing, the intervals of adjacent k z1 are unequal. The intervals decrease with the increase of k z1 , which differ obviously from the cases of ρ -SO/ z-PR mixing modes (refer to Fig. 1). Moreover, the larger the azimuthal quantum number m, the adjacent intervals of k z1 become smaller. Comparing the dispersive features of z-SO/ρ -PR mixing modes in GaN QD with those of the phonon modes in GaN QWs [1–4] and QWRs systems [6,7], we find that the z-SO/ρ -PR mixing modes in wurtzite cylindrical QDs possess both characteristics of the SO phonon modes in wurtzite QWs and the PR phonon modes in wurtzite QWRs structures. For example, the SO-PR mixing modes in wurtzite cylindrical QD systems have two branches. One is symmetrical, the other is antisymmetrical, and they converge to a cer-
In the present work, we have investigated the SO-PR mixing phonon modes in a wurtzite cylindrical QD system. It is found that there exist two types of SO-PR mixing phonon modes in wurtzite cylindrical QD structures, namely the ρ -SO/ z-PR mixing modes and the z-SO/ρ -PR mixing modes. And each SO-PR mixing phonon modes also have two forms. One is symmetrical, and the other is antisymmetrical. Based on the method of field quantization, the Fröhlich Hamiltonian of electron–(SO-PR) mixing phonons interactions are given. Numerical calculations on the dispersive properties of the SO-PR mixing phonon modes of a wurtzite GaN cylindrical QD are performed. We observed that the dispersive spectra of SO/PR mixing modes just only take a serial of discrete values due to the three-dimension confinement of the QD structure. As the height or the radius of the Q0D wurtzite cylindrical QDs approach infinity, both types of the SO-PR mixing modes will reduce to the SO modes or PR modes in Q2D wurtzite QW and Q1D QWR structures. These reducing behaviors of the mixing modes have been analyzed in depth from both of the viewpoints of physics and mathematics. This shows the theories of mixing phonon modes in wurtzite Q0D QDs established in the present Letter are consistent with those in wurtzite Q1D QW and Q0D QWR systems [1–4,6,7]. Therefore the present theories and numerical results are correct and reliable. 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 Q0D wurtzite QD systems. References [1] S.M. Komirenko, K.W. Kim, M.A. Stroscio, M. Dutta, Phys. Rev. B 59 (1999) 5013. [2] J. Gleize, M.A. Renucci, J. Frandon, F. Demangeot, Phys. Rev. B 60 (1999) 15985; J. Gleize, M.A. Renucci, J. Frandon, F. Demangeot, Mater. Sci. Eng. B 82 (2001) 27. [3] J.J. Shi, Phys. Rev. B 68 (2003) 165335. [4] L. Zhang, J.J. Shi, Commun. Theor. Phys. 45 (2006) 935. [5] L.T. Tan, R.W. Martin, K.P. O’Donnell, I.M. Watson, Appl. Phys. Lett. 89 (2006) 101910. [6] L. Zhang, J.J. Shi, T.L. Tansley, Phys. Rev. B 71 (2005) 245324. [7] L. Zhang, J.S. Liao, Superlatt. Microstruct. 43 (2008) 28. [8] A.F. Jarjoura, et al., Superlatt. Microstruct. 43 (2008) 431. [9] N. Garro, et al., Phys. Rev. B 74 (2006) 075305. [10] T. Xu, et al., J. Appl. Phys. 102 (2007) 073517. [11] J.J. Shi, T.L. Tansley, Solid State Commun. 138 (2006) 26. [12] L. Liu, J. Li, G. Xiong, Physica E 25 (2005) 466. [13] C.X. Wang, G.G. Xiong, Microelectron. J. 37 (2006) 847. [14] C.X. Xia, F. Jiang, S.Y. Wei, Current Appl. Phys. 8 (2008) 153. [15] V.A. Fonoberov, A.A. Balandin, Phys. Rev. B 70 (2004) 233205. [16] P.M. Chassaing, et al., Phys. Rev. B 77 (2008) 153306. [17] L. Zhang, J.J. Shi, H.J. Xie, Solid State Commun. 140 (2006) 549. [18] R. Loudon, Adv. Phys. 13 (1964) 423. [19] J.Y. Zeng, Quantum Mechanics, second ed., Science Press, Beijing, 2000, pp. 744–750 (in Chinese).