Applying förster-type nonradiative energy transfer formalism to nanostructures with various directionalities : dipole electric potential of exciton and dielectric environment

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(1)Chapter 1. Applying Förster-Type Nonradiative Energy Transfer Formalism to Nanostructures with Various Directionalities: Dipole Electric Potential of Exciton and Dielectric Environment. In this chapter, we present analytical equations for the exciton electric potential inside and outside a nanostructure; including analytical expressions, for the long distance approximation, which are derived for the outside electric potential. Finally, the effective dielectric constant expressions, for this limit, are obtained. This chapter is reprinted (adapted) with permission from Ref. [1]. Copyright 2013 American Chemical Society.. 1.1. Spherical Geometry: Nanoparticle Case. The electric potential for an exciton in the a-direction ða ¼ x; y; zÞ, illustrated in Fig. 1.1a, is given by Uin a ¼. ^r edexc a 2ðeNP e0 Þ r 3 1 þ eNP þ 2e0 R3NP eNP r 3. ð1:1Þ. edexc 3eNP r^ a eNP eNP þ 2e0 r 3. ð1:2Þ. Uout a ¼. where eNP and e0 are the nanoparticle (NP) and medium dielectric constants, respectively. The electric potential is the same in any direction because of the spherical symmetry of the NP. In the long distance approximation the outside electric potential can be written as. © The Author(s) 2017 P.L. Hernández Martínez et al., Understanding and Modeling Förster-type Resonance Energy Transfer (FRET), Nanoscience and Nanotechnology, DOI 10.1007/978-981-10-1873-2_1. 1.

(2) 2. Applying Förster-Type Nonradiative Energy …. 1. Uout a. edexc r^ a ¼ r3 eeff. ð1:3Þ. where eeff is the effective dielectric constant given by eeff ¼. 1.2. eNP þ 2e0 3. ð1:4Þ. Cylindrical Geometry: Nanowire Case. In this case, the electric potential for an a-exciton ða ¼ x; y; zÞ, illustrated in Fig. 1.1a, is. (a). Total Electric Potential NP (a.u). (b) 5.0x10-5 4.0x10. Total Electric Potential NP (a.u) Outside Electric Potential NP (a.u). -5. 3.0x10-5 2.0x10-5 1.0x10-5 0.0 0. 1. 2. 3. 4. 5. 6. z (nm). (d) Total Electric Potential QW (a.u). Total Electric Potential NW (a.u). (c) Total Electric Potential NW (a.u) Outside Electric Potential NW (a.u). 5.0x10-5 4.0x10-5 3.0x10-5 2.0x10-5 1.0x10-5 0.0 0. 1. 2. 3. z (nm). 4. 5. 6. 5.0x10-5. Total Electric Potential QW (a.u) Outside Electric Potential QW (a.u). 4.0x10-5 3.0x10-5 2.0x10-5 1.0x10-5 0.0 0. 5. 10. 15. 20. 25. z (nm). Fig. 1.1 a Schematic of an exciton in an NP, an NW, and a QW. Red circle represents an exciton in the a-direction. RNP(NW) is the NP (NW) radius. LQW is the QW capping layer thickness. b, c, and d Electric potential along the “z” axis for a z-exciton. Total and long distance approximation electric potential for the z-exciton inside: b an NP; c an NW; and d a QW [Reprinted (adapted) with permission from Ref. [1] (Copyright 2013 American Chemical Society)].

(3) 1.2 Cylindrical Geometry: Nanowire Case. Uin a. 3. XZ eimu eiky Aam ðkÞIm ðjk j qÞ dk ¼ Ua þ. ð1:5Þ. m. Uout a. XZ . ¼ Ua þ. eimu eiky Bam ðkÞKm ðjk j qÞ dk. ð1:6Þ. m. where Im ðjk j qÞ and Km ðjk j qÞ are the modified Bessel functions of order m, and Ua is the a-exciton electric potential. After applying the boundary conditions at the surface of the nanowire (NW), the coefficients Aam and Bam are Aam ðk Þ ¼ Bam ðk Þ ¼. eNW. . Km ðjk j RNW Þ a B m ðk Þ Im ðjk j RNW Þ. ðe0 eNW Þgam ðjkjÞ Km ðjk j RNW Þ Im ðjk j RNW Þ I m ðjk j RNW Þ þ e0 K m ðjk j RNW Þ 2 jk j. ð1:7Þ ð1:8Þ. where I m ðjk j RNW Þ; K m ðjk j RNW Þ, and gam ðjkjÞ are defined as I m ðjk j RNW Þ ¼ Im1 ðjk j RNW Þ þ Im þ 1 ðjk j RNW Þ. ð1:9Þ. K m ðjk j RNW Þ ¼ Km1 ðjk j RNW Þ þ Km þ 1 ðjk j RNW Þ. ð1:10Þ. . @Ua e @q q¼RNW. ð1:11Þ. gam ðjkjÞ. ¼. 1 ð2pÞ2. Z2p Z1 dudye 0. 1. imu iky. For an exciton in the y-direction (along the cylinder axis), the coefficient Bym becomes By0 ðkÞ. 0 1 edexc i @ 1 A ¼ ðeNW e0 Þ jkj K0 ðjk jRNW ÞI1 ðjk jRNW Þ p eNW eNW þ e0. ð1:12Þ. K1 ðjk jRNW ÞI0 ðjk jRNW Þ. with an electric potential given by Uout y. Z iky y edexc y ¼ e B0 ðk ÞK0 ðjk j qÞ dk 3 þ 2 2 2 eNW ðq þ y Þ. ð1:13Þ. In the long distance approximation, the coefficient Bym and the outside electric potential are simplified as.

(4) 4. Applying Förster-Type Nonradiative Energy …. 1. By0 ðk Þ. edexc i 1 ¼ ðeNW e0 Þ jk j p e0 eNW edexc y out Uy ¼ 3 eeff ðq2 þ y2 Þ2. ð1:14Þ ð1:15Þ. where eeff is the effective dielectric constant defined as eeff ¼ e0. ð1:16Þ. In the case of an exciton in the z-direction (perpendicular to the cylinder axis), the coefficient Bzm which remains as Bz1 and Bz1 , where Bz1 is given . Bz1 ðkÞ ¼. . jkjR 2 1. NW 2 2. 0; 1; 1 K2 ðjkjRNW Þ. ðjk j RNW Þ 2 edexc jk j 4 ðK0 ðjkjRNW Þ þ K2 ðjkjRNWÞÞ ðe0 eNW Þ 2p 3 K1 ðjkjRNW Þ I0 ðjkjRNW Þ þ I2 ðjkjRNW Þ eNW þ e0 eNW 1. I1 ðjk jRNW Þ. 2;1 2 G1;3. ð1:17Þ. K0 ðjk jRNW Þ þ K2 ðjk jRNW Þ. 1 1. . 2 A is the Meijer G-function and Bz ¼ Bz . The electric and 1 1. 0; 1; 1 2 potential is simplified as 0. @ jkjRNW 2. G2;1 1;3 ð 2 Þ. Uout z ¼. Z iky z edexc q cosðuÞ þ 2 cos ð u Þ e B1 ðkÞK1 ðjk j qÞ dk 3 eNW ðq2 þ y2 Þ2. ð1:18Þ. In the long distance approximation, the coefficient B and the electric potential become Bz1 ðkÞ ¼ . edexc 1 1 ðe0 eNW Þ jkj 2p eNW þ e0 eNW edexc q cos ðuÞ Uout ¼ 3 z eeff ðq2 þ y2 Þ2. ð1:19Þ ð1:20Þ. where eeff is the effective dielectric constant defined as eeff ¼. eNW þ e0 2. ð1:21Þ. Similarly, for an exciton in the x-direction (perpendicular to the cylinder axis), the non-zero coefficients are Bx1 and Bx1 , where Bx1 is given by.

(5) 1.2 Cylindrical Geometry: Nanowire Case. 5 . Bx1 ðk Þ. jkjR 2 1. NW 2 K ð k Þ j jR 2 NW 2. 0; 1; 1. ðjk j RNW Þ 2 edexc jk j 4 ðK0 ðjkjRNW Þ þ K2 ðjkjRNWÞÞ ¼ ðe0 eNW ÞðiÞ ð1:22Þ 2p 3 K1 ðjkjRNW Þ I0 ðjkjRNW Þ þ I2 ðjkjRNW Þ eNW þ e0 eNW 1. I1 ðjk jRNW Þ. 2;1 2 G1;3. K0 ðjk jRNW Þ þ K2 ðjk jRNW Þ. with Bx1 ¼ Bx1 and the electric potential Uout x ¼. Z iky x edexc q sinðuÞ þ i2 sin ð u Þ e B1 ðkÞK1 ðjk j qÞ dk 3 eNW ðq2 þ y2 Þ2. ð1:23Þ. the coefficients B and the outside electric potential, in the long distance approximation, are simplified as Bx1 ðk Þ ¼. edexc i 1 ðe0 eNW Þ jk j 2p eNW þ e0 eNW edexc q sin ðuÞ Uout ¼ 3 x eeff ðq2 þ y2 Þ2. ð1:24Þ ð1:25Þ. where eeff is the effective dielectric constant, which is defined as eeff ¼. 1.3. eNW þ e0 2. ð1:26Þ. Planar Geometry: Quantum Well Case. The electric potential, in cylindrical coordinates, for an a-exciton ða ¼ x; y; zÞ, illustrated in Fig. 1.1a, is XZ. 1. Uin a. ¼ Ua þ. m. kdkeim/ Jm ðkqÞAam ðk Þ cosh ðkzÞ. ð1:27Þ. kdkeim/ Jm ðkqÞBam ðk ÞExpðkjzjÞ. ð1:28Þ. 0. XZ. 1. Uout a. ¼ Ua þ. m. 0. where Jm ðkqÞ is the Bessel function of order m, and Ua is the a-exciton electric potential. After applying the boundary conditions at the surface of the QW, the coefficients Aam and Bam are.

(6) 6. 1. Aam ðkÞ Bam ðkÞ ¼. Applying Förster-Type Nonradiative Energy …. exp ðjk j LQW Þ a ¼ Bm ðk Þ cosh ðjk j LQW Þ. ðe0 eQW Þham ðjkjÞ kðeQW tanh ðjk j LQW Þ þ e0 Þejk j LQW. ð1:29Þ ð1:30Þ. where ham ðjkjÞ is defined as ham ðjk jÞ. 1 ¼ ð2pÞ. Z2p Z1 duqdqeimu Jm ðkqÞ 0. 0. . @Ua @z z¼LQW. ð1:31Þ. For an exciton in the z-direction, the non-zero coefficient is Bz0 ðkÞ. ¼. edexc ðeQW e0 Þ eQW ðeQW tanh ðkLQW Þ þ e0 Þ. ð1:32Þ. and the electric potential is Uout z. ¼. Z1 edexc z kdkJ0 ðkqÞBz0 ðk ÞExpðkjzjÞ 3 þ eQW ðq2 þ z2 Þ2. ð1:33Þ. 0. Thus, in the long distance approximation, the coefficient B and the electric potential are simplified as Bz0 ðkÞ Uout z ¼. edexc ðeQW e0 Þ e0 eQW. ð1:34Þ. edexc z 3 2 eeff ðq þ z2 Þ2. ð1:35Þ. where eeff is the effective dielectric constant defined as eeff ¼ e0. ð1:36Þ. In the case of an exciton in the x-direction, the non-zero B coefficients are Bx1 ðkÞ and Bx1 ðk Þ, where Bx1 ðkÞ ¼ Bx1 ðkÞ and Bx1 ðk Þ. 1 edexc ðeQW e0 Þ ¼ 2 eQW ðeQW tanh ðkLQW Þ þ e0 Þ. ð1:37Þ.

(7) 1.3 Planar Geometry: Quantum Well Case. 7. the outside electric potential is Uout x. Z1 edexc q cos ð/Þ ¼ kdkJ1 ðkqÞBx1 ðkÞExpðkjzjÞ 3 þ 2 cos ð/Þ eQW ðq2 þ z2 Þ2. ð1:38Þ. 0. In the long distance approximation, the coefficient B and the electric potential are simplified into Bx0 ðk Þ . 1 edexc ðeQW e0 Þ 2 eQW e0 . Uout x ¼. edexc q cos ð/Þ 3 eeff ðq2 þ z2 Þ2. ð1:39Þ ð1:40Þ. where eeff is the effective dielectric constant defined as eeff ¼ e0. ð1:41Þ. Similarly, for an exciton in the y-direction, the non-zero B coefficients are By1 ðkÞ and By1 ðk Þ, where By1 ðkÞ ¼ By1 ðkÞ and By1 ðk Þ ¼. i edexc ðeQW e0 Þ 2 eQW ðeQW tanh ðkLQW Þ þ e0 Þ. ð1:42Þ. with the electric potential given by Uout y. Z1 edexc q sin ð/Þ ¼ kdkJ1 ðkqÞBy1 ðkÞExpðk jzjÞ 3 i2 sin ð/Þ eQW ðq2 þ z2 Þ2. ð1:43Þ. 0. Thus, in the long distance approximation, the coefficient B and the outside electric potential are i edexc ðeQW e0 Þ ð1:44Þ By1 ðk Þ 2 eQW e0 Uout y ¼. edexc q sin ð/Þ 3 eeff ðq2 þ z2 Þ2. ð1:45Þ. where eeff is the effective dielectric constant defined as eeff ¼ e0. ð1:46Þ.

(8) 8 Table 1.1 Effective dielectric constant expressions for NP, NW, and QW cases in the long distance approximation. 1. Applying Förster-Type Nonradiative Energy …. a-direction. NP. NW. QW. x. eeff ¼ eNP þ3 2e0. eeff ¼ eNW 2þ e0. eeff ¼ e0. y. eeff ¼ eNP þ3 2e0. eeff ¼ e0. eeff ¼ e0. z. eNP þ 2e0 3. eeff ¼. eeff ¼. eNW þ e0 2. eeff ¼ e0. This table follows the geometries given in Fig. 1.1 [Reprinted (adapted) with permission from Ref. [1] (Copyright 2013 American Chemical Society)]. A summary for the effective dielectric constant, for the long distance approximation, is given in Table 1.1. Table 1.1 shows the screening factor in the electric potential for different confinement geometries, which corresponds to the NP, NW, and QW cases. This screening factor comes from the boundaries conditions of the electric potential at the interface between the nanostructure (NP, NW, and QW) and the medium. For example, the screening factor for the NP case is the same for an exciton in the x-, y- and z-direction because of its spherical symmetry. In the cylindrical symmetry (NW case), an exciton in the cylindrical main axis does not have any screening factor. However, an exciton perpendicular to the cylindrical main axis has a screening factor as shown in Table 1.1. In the QW case, the screening factor is the same for the x-, y- and z-direction because the QW was considered infinitesimal thin. Table 1.1 follows the geometries sketched in Fig. 1.1a. Figure 1.1 depicts the total and long distance approximation electric potentials for a z-exciton along the z axis. Figure 1.1b shows electric potentials in both the total and long distance approximation for a z-exciton inside an NP. It can be observed that both electric potentials overlap with each other because of the spherical symmetry of the NP nanostructure. The total and long distance approximation electric potentials for a z-exciton in a NW are depicted in Fig. 1.1c. In close proximity to the NW surface, the long distance approximation underestimates the exciton electric potential, as it is shown in Fig. 1.1c. In the QW case, the long distance approximation overestimates the exciton electric potential in the close proximity to the QW surface (Fig. 1.1d). This is an opposite effect compared to the NW case. These underestimation and overestimation of the electric potential, for NW and QW, respectively, is due to the fact that at short distances the long distance approximation do not apply and higher effects need to be considered. However, in all cases, at long distances the total electric potential converges into the long distance approximation (Fig. 1.1b–d).. Reference 1. P.L. Hernández-Martínez, A.O. Govorov, H.V. Demir, Generalized theory of Förster-type nonradiative energy transfer in nanostructures with mixed dimensionality. J. Phys. Chem. C 117, 10203–10212 (2013).

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