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Nonlocal response of size-quantized excitons in a semiconductor microsphere
JOURNALO~
——
LUMINESCENCE
ELSEVIER
Journal of Luminescence ô0&ô1
994) 33)) 312
Nonlocal response of size-quantized excitons in a semiconductor microsphere K. Cho*. M. Nishida, Y. Ohfuti, L. Belleguie Io(nhIi
of L’ngnucrme 2)rien(c ().soka
nhuer,s ia
56(1 7~,i,~uikoJapan
Abstract By using nonlocal response theory which determines the selfconsistent motions of induced polarization and radiation held, the linear response of size-quantized excitons in a semiconductor microsphere has been calculated as a function of the radius of the sphere. The induced polarization for resonant light shows a remarkable size-resonant enhancement.
It has recently become very popular to study mesoscopic (MS) systems. One of the most remarkable points about MS systems is the appearance of quantum mechanical coherence in observed quantities. In the case of optical response. it manifests itself as size quantization of the material levels. This leads to a size enhancement of oscillator strengths of the excited states [1], which has attracted much attention from applicational points of view. However, this is not the only aspect of material coherence. The consideration of the wave functions in detail, which leads to the condition of size qLiantiration, also requires a particular nonlocal relationship between the held and the polari7ation. The nonlocal form of the susceptibility is the other aspect to be considered. One of the present authors has formulated a nonlocal response theory [2] which solves the Maxwell equations for a microscopic radiation held and the Schrodinger equation for matter system selfconsistently. This leads to a set of integro-differential
*
Corresponding author.
equations to determine the vector potential A (or the source electric held E~)and the current density j (or the induced polarization P). Using the separable character of the integral kernel giving j as a functional of A, we can reduce the siniultaneous integro-differential equations. in the case of linear response, to a set of linear equations of the form -
(L,•
0 =
—-
ho~
C.
(L~ + hw =
i()
-
)X ~ + ~ ~(D)VX~ ~ D~,A0,
,
+ 10 ) .Y~ +
~
.V~+ U,~V0.)
(‘~. ..
where L -° is the ,.th excitation energy of the matter , system. the D s arc the retarded interactions (via photons) between the various excited states, and the ( s represent the (known) amplitudes of the incident held. In terms of the X s of these equations, . the induced polarization is expanded as
P(r. w)
0022-2313;94, SOlO)) ‘, 1994 Elsevier Science B.V. All rights resersed ‘~SI)l0O22-23l3)93)E()277-3
=
~S[X,0p0~ Ii’)
+
X,p~o(r)1.
(3)
K. Cho et al. / Journal of Lununescence 60&61 (1994) 330—332
331
where pv~(r)is the density of the transition dipole moment between the states ~.,v). Since the response field is calculated from this P(r, w), it is also a linear function of the X’s. It should be noted that the solution of X of (1) and (2) contains the effect of radiative shift and broadening because of the presence of the retarded interaction in these equations. For the cases of a single two-level atom (see e.g. [3]) and 1D and 2D periodic arrays of them [4], where the calculations in QED (Quantum Electrodynamics) are available, the above equations are shown to give the same expressions for the radiative broadening as QED [5]. Thus, we can expect that this semiclassical formalism can correctly deal
to screen Hdd by a background dielectric constant in order to work with a finite number of resonant states.) In terms of the eigenfunctions (spherical Bessel functions multiplied with spherical harmonics Yr) of KCM, it is easily shown that KCM is proportional to R 2, and the eigenvalues of Had are R-independent. Taking an appropriate number of basis functions (four lowest n’s for / = 0 and / = 2 in the following figures), we have solved Eqs. (1) and (2), and calculated the response spectrum (the intensity of the radiation field at infinity elastically scattered from the sphere) as in Fig. 1. The material parameters are those for the CuC1[Z3] exciton given in Ref. [5].
with the radiative effect in general. [This expectation can be confirmed by the comparison of the equations of motion for matter excitation and a radiation field in QED with the above Eqs. (1) and (2).] An extension of this formalism to nonlinear response is also possible, and several examples have already been treated by such a nonlocal nonlinear response theory [6]. Among the various possible cases of application, we consider tn this paper the case of linear response of size-quantized excitons in a fine semiconducting particle. This study is interesting because fine particles are one of the most typical mesoscopic systems, and because a fully nonlocal treatment of a single fine particle has never been performed (except for a treatment by Ruppin based on a polariton picture with assumed additional boundary conditions [7]). We consider a sphere of radius R which is much larger than the radius of the electron—hole relative motion. In such a weak confinement regime, only the center of mass (CM) motion of the exciton is size quantized. The quantum numbers (n,1, m) of the exciton correspond to the radial (n) and angular (l,m) parts of the CM motion in a sphere. The matter Hamiltonian determining the excitation energies {E~0} in Eqs. (1) and (2) consists of the size-quantized kinetic energy of the CM motion KCM and the instantaneous interaction Hdd between the induced dipole distributions. (In the present nonlocal formulation, it is essential to explicitly define the gauge and we have chosen the Coloumb gauge. For a different choice of gauge, ~ may be ascribed to the depolarization shift. It is necessary
Fig. 2 shows the R-dependence of the expansion coefficients {X} for resonant light at the lowest size-quantized level (hw = E~02 = 1), which demonstrates a remarkable size resonance. This behavbr of the expansion coefficients X can be interpreted as a consequence of large radiative shifts of {E,~}as a function of R. Let us denote the coefficient matrix on the left hand side of Eqs. (1) and (2) as S. and suppose that det [S] = 0 is satisfied by the complex frequencies {w = Q2}. In the absence of the retarded interaction, the numbering of Q~is
.
(a)
R=300 (A)
(b)
R=500 (A)
\J _____._—~
___________________
3. 20 .
3. 205 .
.
.
(eV)
3. 21
Fig. 1. Intensity (in arbitrary units) of elastically scattered light at infinity for (a) R = 300 A and (b) R = 500 A. The dashed lines represent {E50}, the excitation energies of the matter.
332
K.
(ho or ii.
Journal o/ Lunii,in’.crn’,ins’ 60&6J
H
2
•1 -
II
500
0. 3.
000
Fig. 2. [he R-dependences of the expansion coethcientsH\~
5 in arbitrary units. Each peak corresponds to a NI DORES coil-
1994
?~) 11’
denote this effect as “nonlocality induced double resonance in energy and size” (NIDORES). This effect was found for the first time for a slab [8]. and the present result for a sphere is the second example of NIDORES. Since the above explanation does not depend on a particular model, we can generally expect the NIDORES effect in most MS systems. Because of the resonance enhancement of a particular spatial pattern in the induced polarization. one can expect an exciting nonlinear effect when a pump beam satisfies a NI DORES condition [9].
di ion. (See text.)
such that hc2~= E~.Due to the retarded interaction D, D, etc.. every ~ is shifted by Re[Q~] E,,0. It is a characteristic point of MS systems that this shift can well exceed the interval of neighboring EAO’s. If we increase R by keeping the (frequency) resonance condition w J(~o/h,a series of resonance E~0 Re[Q5] Li 0(t’ I) may occur for particular values of R. At each resonance there is a very small factor Im [Q~.] in the product det [S] = I I~(w Q~).This leads to a sizeresonant enhancement of the solution X~0.Therefore, the induced polarization has a very strong component of the spatial pattern represented by Pu ,(r) at the resonance, in spite of the fact that the frequency resonance is kept at L~0. Since all aspects of this resonance are essentially related with the nonlocal nature of the medium, we =
—
=
—
—
This work was partially supported by a Grantin-aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. References 1 1
E Hanamura. Phys Rex B 37 (98(1)
273
[2] K. (‘ho. Prog. [heor. Phys. Suppl. 06 (1991) 225. 13] H. Haken, laser Theory (Springer, Berlin. 970) Section V.3. 141 M. Orrit. C ..Aslangul and P. Kottice, Phys. Rev. B 25)19(1)0 [~1 Y Ohfuti and K Cho, Jpn J Appl Phys 9 (19931 97 ~1 H Ishihar~iand K (‘ho. J. Nonlin Opt Phys. 1(1992)287: Phys. Rev. B, in print. [7] R. Ruppin. J. Phys. Chem. Solids 501989)877. [8] H. Ishihara and K. Cho. in: Proc. Ini. Symp. on Science and Technology of Mesoscopic Structures. eds. 5. Namba. (‘. Hamaguehi and T Ando (Springer. Berlin. 1992) p 457
[9] H lshihara and K (‘ho. QELS Technical Digest Series 12 (1993) 82.
——
LUMINESCENCE
ELSEVIER
Journal of Luminescence ô0&ô1
994) 33)) 312
Nonlocal response of size-quantized excitons in a semiconductor microsphere K. Cho*. M. Nishida, Y. Ohfuti, L. Belleguie Io(nhIi
of L’ngnucrme 2)rien(c ().soka
nhuer,s ia
56(1 7~,i,~uikoJapan
Abstract By using nonlocal response theory which determines the selfconsistent motions of induced polarization and radiation held, the linear response of size-quantized excitons in a semiconductor microsphere has been calculated as a function of the radius of the sphere. The induced polarization for resonant light shows a remarkable size-resonant enhancement.
It has recently become very popular to study mesoscopic (MS) systems. One of the most remarkable points about MS systems is the appearance of quantum mechanical coherence in observed quantities. In the case of optical response. it manifests itself as size quantization of the material levels. This leads to a size enhancement of oscillator strengths of the excited states [1], which has attracted much attention from applicational points of view. However, this is not the only aspect of material coherence. The consideration of the wave functions in detail, which leads to the condition of size qLiantiration, also requires a particular nonlocal relationship between the held and the polari7ation. The nonlocal form of the susceptibility is the other aspect to be considered. One of the present authors has formulated a nonlocal response theory [2] which solves the Maxwell equations for a microscopic radiation held and the Schrodinger equation for matter system selfconsistently. This leads to a set of integro-differential
*
Corresponding author.
equations to determine the vector potential A (or the source electric held E~)and the current density j (or the induced polarization P). Using the separable character of the integral kernel giving j as a functional of A, we can reduce the siniultaneous integro-differential equations. in the case of linear response, to a set of linear equations of the form -
(L,•
0 =
—-
ho~
C.
(L~ + hw =
i()
-
)X ~ + ~ ~(D)VX~ ~ D~,A0,
,
+ 10 ) .Y~ +
~
.V~+ U,~V0.)
(‘~. ..
where L -° is the ,.th excitation energy of the matter , system. the D s arc the retarded interactions (via photons) between the various excited states, and the ( s represent the (known) amplitudes of the incident held. In terms of the X s of these equations, . the induced polarization is expanded as
P(r. w)
0022-2313;94, SOlO)) ‘, 1994 Elsevier Science B.V. All rights resersed ‘~SI)l0O22-23l3)93)E()277-3
=
~S[X,0p0~ Ii’)
+
X,p~o(r)1.
(3)
K. Cho et al. / Journal of Lununescence 60&61 (1994) 330—332
331
where pv~(r)is the density of the transition dipole moment between the states ~.,v). Since the response field is calculated from this P(r, w), it is also a linear function of the X’s. It should be noted that the solution of X of (1) and (2) contains the effect of radiative shift and broadening because of the presence of the retarded interaction in these equations. For the cases of a single two-level atom (see e.g. [3]) and 1D and 2D periodic arrays of them [4], where the calculations in QED (Quantum Electrodynamics) are available, the above equations are shown to give the same expressions for the radiative broadening as QED [5]. Thus, we can expect that this semiclassical formalism can correctly deal
to screen Hdd by a background dielectric constant in order to work with a finite number of resonant states.) In terms of the eigenfunctions (spherical Bessel functions multiplied with spherical harmonics Yr) of KCM, it is easily shown that KCM is proportional to R 2, and the eigenvalues of Had are R-independent. Taking an appropriate number of basis functions (four lowest n’s for / = 0 and / = 2 in the following figures), we have solved Eqs. (1) and (2), and calculated the response spectrum (the intensity of the radiation field at infinity elastically scattered from the sphere) as in Fig. 1. The material parameters are those for the CuC1[Z3] exciton given in Ref. [5].
with the radiative effect in general. [This expectation can be confirmed by the comparison of the equations of motion for matter excitation and a radiation field in QED with the above Eqs. (1) and (2).] An extension of this formalism to nonlinear response is also possible, and several examples have already been treated by such a nonlocal nonlinear response theory [6]. Among the various possible cases of application, we consider tn this paper the case of linear response of size-quantized excitons in a fine semiconducting particle. This study is interesting because fine particles are one of the most typical mesoscopic systems, and because a fully nonlocal treatment of a single fine particle has never been performed (except for a treatment by Ruppin based on a polariton picture with assumed additional boundary conditions [7]). We consider a sphere of radius R which is much larger than the radius of the electron—hole relative motion. In such a weak confinement regime, only the center of mass (CM) motion of the exciton is size quantized. The quantum numbers (n,1, m) of the exciton correspond to the radial (n) and angular (l,m) parts of the CM motion in a sphere. The matter Hamiltonian determining the excitation energies {E~0} in Eqs. (1) and (2) consists of the size-quantized kinetic energy of the CM motion KCM and the instantaneous interaction Hdd between the induced dipole distributions. (In the present nonlocal formulation, it is essential to explicitly define the gauge and we have chosen the Coloumb gauge. For a different choice of gauge, ~ may be ascribed to the depolarization shift. It is necessary
Fig. 2 shows the R-dependence of the expansion coefficients {X} for resonant light at the lowest size-quantized level (hw = E~02 = 1), which demonstrates a remarkable size resonance. This behavbr of the expansion coefficients X can be interpreted as a consequence of large radiative shifts of {E,~}as a function of R. Let us denote the coefficient matrix on the left hand side of Eqs. (1) and (2) as S. and suppose that det [S] = 0 is satisfied by the complex frequencies {w = Q2}. In the absence of the retarded interaction, the numbering of Q~is
.
(a)
R=300 (A)
(b)
R=500 (A)
\J _____._—~
___________________
3. 20 .
3. 205 .
.
.
(eV)
3. 21
Fig. 1. Intensity (in arbitrary units) of elastically scattered light at infinity for (a) R = 300 A and (b) R = 500 A. The dashed lines represent {E50}, the excitation energies of the matter.
332
K.
(ho or ii.
Journal o/ Lunii,in’.crn’,ins’ 60&6J
H
2
•1 -
II
500
0. 3.
000
Fig. 2. [he R-dependences of the expansion coethcientsH\~
5 in arbitrary units. Each peak corresponds to a NI DORES coil-
1994
?~) 11’
denote this effect as “nonlocality induced double resonance in energy and size” (NIDORES). This effect was found for the first time for a slab [8]. and the present result for a sphere is the second example of NIDORES. Since the above explanation does not depend on a particular model, we can generally expect the NIDORES effect in most MS systems. Because of the resonance enhancement of a particular spatial pattern in the induced polarization. one can expect an exciting nonlinear effect when a pump beam satisfies a NI DORES condition [9].
di ion. (See text.)
such that hc2~= E~.Due to the retarded interaction D, D, etc.. every ~ is shifted by Re[Q~] E,,0. It is a characteristic point of MS systems that this shift can well exceed the interval of neighboring EAO’s. If we increase R by keeping the (frequency) resonance condition w J(~o/h,a series of resonance E~0 Re[Q5] Li 0(t’ I) may occur for particular values of R. At each resonance there is a very small factor Im [Q~.] in the product det [S] = I I~(w Q~).This leads to a sizeresonant enhancement of the solution X~0.Therefore, the induced polarization has a very strong component of the spatial pattern represented by Pu ,(r) at the resonance, in spite of the fact that the frequency resonance is kept at L~0. Since all aspects of this resonance are essentially related with the nonlocal nature of the medium, we =
—
=
—
—
This work was partially supported by a Grantin-aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. References 1 1
E Hanamura. Phys Rex B 37 (98(1)
273
[2] K. (‘ho. Prog. [heor. Phys. Suppl. 06 (1991) 225. 13] H. Haken, laser Theory (Springer, Berlin. 970) Section V.3. 141 M. Orrit. C ..Aslangul and P. Kottice, Phys. Rev. B 25)19(1)0 [~1 Y Ohfuti and K Cho, Jpn J Appl Phys 9 (19931 97 ~1 H Ishihar~iand K (‘ho. J. Nonlin Opt Phys. 1(1992)287: Phys. Rev. B, in print. [7] R. Ruppin. J. Phys. Chem. Solids 501989)877. [8] H. Ishihara and K. Cho. in: Proc. Ini. Symp. on Science and Technology of Mesoscopic Structures. eds. 5. Namba. (‘. Hamaguehi and T Ando (Springer. Berlin. 1992) p 457
[9] H lshihara and K (‘ho. QELS Technical Digest Series 12 (1993) 82.