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Phonon induced pure dephasing process of excitonic state in colloidal semiconductor quantum dots
Superlattices and Microstructures 92 (2016) 52e59
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Phonon induced pure dephasing process of excitonic state in colloidal semiconductor quantum dots Tongyun Huang, Peng Han*, Xinke Wang, Shengfei Feng, Wenfeng Sun, Jiasheng Ye, Yan Zhang** Department of Physics, Beijing Key Lab for Metamaterials and Devices and Key Laboratory of Terahertz Optoelectronics Ministry of Education, Capital Normal University, Beijing 100048, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 January 2016 Accepted 29 January 2016 Available online 8 February 2016
We present a theoretical study on the pure dephasing process of colloidal semiconductor quantum dots induced by lattice vibrations using continuum model calculations. By solving the time dependent Liouville-von Neumann equation, we present the ultrafast Rabi oscillations between excitonic state and virtual state via exciton-phonon interaction and obtain the pure dephasing time from the fast decayed envelope of the Rabi oscillations. The interaction between exciton and longitudinal optical phonon vibration is found to dominate the pure dephasing process and the dephasing time increases nonlinearly with the reduction of exciton-phonon coupling strength. We further find that the pure dephasing time of large quantum dots is more sensitive to temperature than small quantum dots. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Pure dephasing Semiconductor nanocrystals Ultrafast process
1. Introduction Research on colloidal semiconductor quantum dots (QDs) or called as nanoclusters (NCs) is a rapidly growing field driven by their size tunable electronic and optical properties for the applications as next generation optoelectronics, spintronics, photovoltaics and biolabeling devices [1e7]. Indeed, these applications are usually performed at room temperature where lattice vibrations (phonons) and their interactions with electronic states are naturally involved. In semiconductor QDs, the excited state can be relaxed to the ground state with energy loss to heat via inelastic exciton-phonon scattering, while the phase of the electronic wavefunction is randomized via elastic exciton-phonon interaction and results in loss of quantum coherence, i.e. pure dephasing. The pure dephasing process of QDs is schematically illustrated as a three-level system in Fig. 1. As shown in this figure, the QD is excited from the ground state jg〉 to the excitonic state je1 〉 by a pump pulse. Shortly after the excitation, the system relaxes to state je2 〉 at the bottom of the energy surface via emission phonons. Due to the large energy gap between the excited state je2 〉 and phonon energy, the energy of je2 〉 state is merely shifted slightly without transfer to the other states via coupling to lattice vibrations. During the interaction between excitonic state je2 〉 and lattice vibrations, the phase of the excitonic state wavefunction is destroyed and the quantum coherence of the excited state is lost. In the real applications of nanoscale optoelectronics, photovoltaics, and biological devices, a long-lived quantum coherence of the excited state is partly important for their role in light-harvesting function [8,9]. Moreover, a long-lived quantum coherence is
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (P. Han), [email protected] (Y. Zhang). http://dx.doi.org/10.1016/j.spmi.2016.01.042 0749-6036/© 2016 Elsevier Ltd. All rights reserved.
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
53
Fig. 1. Schematic of the phonon induced pure dephasing process of the excitonic state.
also necessarily needed for the realization of single-photon sources, entangled photons sources, quantum gates and registers in the area of quantum information science and technology. To investigate the pure dephasing process of semiconductor QDs, considerable attentions have been paid from theoretical aspect since the beginning of this century. Based on the definition of quantum decoherence, the dephasing rate of semiconductor self-assemble QDs was firstly calculated via solving the dipole-dipole auto-correlation function using model calculations [10e12]. Later, master equations were applied to study the ultrafast dephasing process of self-assemble QD systems under the excitation of laser field [13,14]. In addition to the model calculations, ab initio time-domain density functional theory (TDDFT) was recently proposed to study the pure dephasing process of semiconductor nanostructures composed by tens to hundreds of atoms [9,15e17]. Although the pure dephasing processes of semiconductor QDs and their effects on the zero phonon line width, Mollow sideband dephasing, and the change of absorption line shape have been studied in the previous researches, a systematical study on the pure dephasing processes of spherical colloidal QDs via exciton acoustic and optical phonon scattering and as a function of QD sizes and temperature is still lacking. In this work, we study the ultrafast pure dephasing process of GaAs, GaP, InP, CdTe, ZnSe and ZnTe spherical colloidal QDs with radius from 3 to 10 nm induced by exciton-phonon interaction via continuum model calculations. The excitonic states of QDs are obtained from effective mass approximation and the exciton-phonon coupling strength are calculated using the €hlich interaction for optical phonon modes. deformation and the piezoelectric potentials for acoustic phonon modes and Fro By solving the Liouville-von Neumann equation with the calculated exciton-phonon coupling strength, the ultrafast pure dephasing process of QDs is obtained. Based on our calculation, we find that: (i) the longitudinal optical (LO) phonon vibrations dominate the pure dephasing process of QDs we studied; (ii) the pure dephasing time increases nonlinear with the decrease of exciton-phonon coupling strength; and (iii) the pure dephasing time of large QDs is more strongly temperature dependent than that of the small dots. 2. Theoretical methods The excitonic state, which is composed by an interacting electron-hole pair, of semiconductor QD with potential barrier Vb is described as,
"
# Z2 2 Z2 2 e2 J ¼ EJ V V þ Vb 2me e 2mh h 4pε0 εr jre rh j
(1)
where, Z denotes the reduced Planck constant, me and mh are the effective masses of electron and hole, ε0 is the vacuum permittivity, εr is the static dielectric constant, e is the electron charge, r e and r h are the positions of the electron and hole, respectively. By solving Eq. (1), the exciton energy E and the corresponding wavefunction of spherical QDs Jnlm ðrÞ ¼ Rnl ðrÞYlm ðq; 4Þ with Rnl ðrÞ the radial function and Ylm ðq; 4Þ the spherical harmonic function, are calculated. The opt optical band gaps of QDs Egap ¼ Ee þ Eh þ Eg E, with Ee the confinement energy of electron, Eh the confinement energy of hole and Eg the band gap of bulk material, are then obtained. In the case of long wavelength acoustic phonons, the atomic displacements correspond to the deformation of lattice and the energy change of the excitonic state induced by the static lattice distortion can be approximated as [25],
54
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
b DF ¼ H
sffiffiffiffiffiffiffiffiffiffiffiffi Z X y VD q$x aq eiq$r þ aq eiq$r 2rVu q
(2)
where V is the volume of QD, r represents the mass density, u is the acoustic phonon frequency, VD denotes the deformation potential induced by static distortion of lattice, q represents the phonon wave vector, x denotes the phonon polarization unit vector, ayq and aq are the creation and the annihilation operators of the acoustic phonon, respectively. In addition to the deformation potential, a stress can also induce a macroscopic electric polarization field in materials without inversion symmetry. The QDs we studied in this work are composed by semiconductors with zinc-blende structure (Td point group symmetry), where only one piezoelectric constant element is independent, d14 ¼ d25 ¼ d36 . The change of excitonic state energy induced by the piezoelectric potential is then written as [26],
b PZ ¼ i H
sffiffiffiffiffiffiffiffiffiffiffiffi Z X y 2ed14 qx qy xz þ qy qz xx þ qz qx xy aq eiq$r þ aq eiq$r 2rVu q
(3)
Combining the deformation and the piezoelectric potentials, the exciton-acoustic phonon interaction matrix Mexph is calculated as [27],
DF b b PZ J〉: Mexph ¼ 〈J H þH
(4)
In addition to the acoustic phonon modes, the relative displacement of oppositely charged atoms, which is induced by the longitudinal optical (LO) phonon vibration in polar materials, generates a macroscopic electric field. The interaction between €hlich interaction with the form [25], this electric field and exciton is known as the Fro
b LO ¼ H
X q
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pe2 ZuLO 1 1 y iq$r aq e þ aq eiq$r 2 ε∞ ε0 εr ε0 q V
(5)
where uLO is the LO phonon frequency and ε∞ is the high frequency dielectric constant. All these calculations are performed using parameters given in Table 1. Once the exciton-phonon coupling matrix elements are obtained, we solve the ultrafast dephasing process of the excited state je2 〉 using the Liouville-von Neumann equation [28,29], Table 1 Material parameters. Material
GaAs
GaP
InP
CdTe
ZnSe
ZnTe
Effective electron mass m*e ðm0 Þ Effective hole mass (heavy) m*hh ðm0 Þ Effective hole mass (light) m*lh ðm0 Þ Potential barrier Vb ðeV Þ Longitudinal sound velocity vl ð105 cm=sÞ Transverse sound velocity vt ð105 cm=sÞ Mass density rðg=cm3 Þ The frequency of LA phonon at X point uLO ðcm1 Þ The frequency of LA phonon at G point uG ðcm1 Þ The frequency of LO phonon uLO ðcm1 Þ Lattice constant að AÞ Static dielectric constant εr high frequency dielectric constant ε∞ Deformation potential for electron VDe ðeV Þ Deformation potential for hole VDh ðeVÞ Piezoelectric constant d14 ð109 V=mÞ Band gap Eg ðeVÞ
0.063a 0.51a 0.082a 4.10c 4.73a 3.35a 5.32a 213g 210g 285b 5.65b 12.9a 10.9b 7.17h 1.16h 1.40a 1.43b
0.090a 0.79a 0.140a 3.58c 5.83a 4.12a 4.14a 246g 233g 403b 5.45b 11.1a 8.8b 8.20h 1.70h 1.02a 2.76b
0.080a 0.60a 0.089a 4.40c 4.58a 3.08a 4.18a 184g 174g 346b 5.87b 12.5a 9.9b 6.00h 0.60h 0.32a 1.35b
0.090b 0.82b 0.145b 4.23c 3.35d 1.79d 5.86f 123g 108g 167b 6.48b 10.2f 7.1b 3.96h 0.55h 0.39i 1.51b
0.137b 0.82b 0.154b 3.99c 4.04e 2.78e 5.26f 184g 167g 252b 5.67b 9.1f 5.9b 4.17h 1.65h 0.61i 2.72b
0.117b 0.67b 0.159b 3.46c 3.55e 2.36e 5.65f 144g 136g 210b 6.01b 7.4f 6.9b 5.83h 0.79h 0.31i 2.27b
a b c d e f g h i
See website: http://www.ioffe.ru/SVA/NSM/Semicond/. Reference [18]. Reference [19]. Reference [20]. Reference [21]. See website: http://www.semiconductors.co.uk/propiivi5410.htm. Reference [22]. Reference [23]. Reference [24].
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
i vb r ðtÞ i hb b ¼ H; r ðtÞ vt Z
55
(6)
The model Hamiltonian of our three-level system is written as [30],
X
b HðtÞ ¼
j¼g;e1 ;e2
1 ðjev 〉〈e2 j þ je2 〉〈ev jÞ þ ZU eiuL t jg〉〈e1 j þ eiuL t je1 〉〈gj Ej jj〉〈jj þ Mexph Nph þ 2
(7)
where Ej represents the eigenvalues of the j th state, jg〉, je1 〉, je2 〉 and jev 〉 are, respectively, the ground state, the excited state, the excitonic state at the bottom of energy surface and the excited virtual state, uL is the frequency of the pulsed pump laser, and U denotes dipole coupling between the light and the system, Nph ¼ 1=expðZuph =kB TÞ 1 is the number of phonon with vibrational frequency uph at temperature with T and kB is the Boltzmann constant. Using the unitary transformation b ¼ exp½iuL t=2ðje 〉〈e j jg〉〈gj þ je 〉〈e jÞ [30] along with the Baker-Hausdorff lemma [31], the Hamiltonian (7) is further U 1 1 2 2 written as,
0
d1 1B b B H ¼ @ ZU 2 0
2ZU d1 2Mexph Nph þ 1=2
1 0 2Mexph Nph þ 1=2 C C A d2
(8)
with energy detuning d1 ¼ Zðue1 ug uL Þ and d2 ¼ d1 þ Zðue2 ue1 Þ, respectively. By solving Eq. (6) with Hamiltonian given in Eq. (8), the ultrafast time dependent dephasing process of the excited state r12 ðtÞ ¼ je1 〉〈e2 j is obtained. 3. Results and discussion The calculated optical band gaps of semiconductor QDs are plotted in Fig. 2 as a function of dot size. We see from this figure that the optical band gap of QDs we studied decreases with dot size increases from 3 to 10 nm and the minimum optical band gap of the QDs we studied (InP QD with 10 nm radium) is around 1.4 eV. Comparing to the phonon energy of QDs (maximum as 50 meV for LO phonon of GaP QD), the excited state cannot be relaxed to the ground state via exciton-phonon coupling due to the large energy detuning and only the coherence of the exciton states losses. Before looking at the pure dephasing process of QDs, we plot the coupling strength of exciton-phonon interaction induced by the LA phonon at the X- and L-points of the Brillouin zone (BZ) along with that induced by the LO phonon around the BZ centre as a function of dot size in Fig. 3. We see from this figure that the strength of exciton-phonon coupling to optical phonon mode is around two orders stronger than that to acoustic phonon mode for QDs we studied. The large exciton-LO phonon coupling is attributed to the strong macroscopic field induced by the relative displacement of cations and anions in the polar IIIeV and IIeVI QDs. Comparing to the strong coupling between exciton and LO phonons, the coupling strength of exciton and LA phonons is much weak for both X- and L-points and increases more rapidly than that to optical phonon mode with decreasing dot size. Once the exciton-phonon coupling strength is obtained, we calculate the pure dephasing process of the excited state using Liouville-von Neumann equation and plot the oscillation process of r12 ðtÞ of GaP QD with radium as 4 nm induced by LO phonon vibrations with time range from 0 to 200 ps in Fig. 4. We see from this figure that the off-diagonal element r12 decays from 0.5 to 0.24 within 200 ps with an exponential envelope function and the dephasing time of the excited state t ¼ 265 ps. To further study the ultrafast dephasing process of the excited state, we plot oscillation of r12 ðtÞ with time from 0 to 0.12 ps as insert in Fig. 4. From this insert figure, we see an ultrafast oscillation between zero to its maximum value within 20 fs. This
Fig. 2. The calculated optical band gap of (a) IIIeV and (b) IIeVI semiconductor QDs as a function of dot size.
56
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
Fig. 3. The size dependent exciton-phonon coupling strength for LA phonon at (a) X-point, (b) L-point, and for (c) LO phonon around the centre of the BZ.
ultrafast oscillation corresponds to the transition between the excited state je2 〉 and the virtual state and the reduction of the oscillation amplitude reflects the loss of coherence. We further plot the pure dephasing time t of GaAs, GaP, InP, CdTe, ZnSe and ZnTe QDs induced by LA phonon at L- and Xpoint and by LO phonon around BZ centre as a function of dot size in Fig. 5 (a)-(f). From this figure, we see that the pure dephasing time induced by acoustic phonon vibration is around few microseconds while that induced by LO phonon is around hundreds of picoseconds for the QDs we studied. Moreover, the pure dephasing time increases with increasing dot size. Both of these characters are determined by the behaviors of exciton-phonon coupling strength, i.e. strong exciton-LOphonon coupling and decrease of exciton-phonon coupling strength with dot size. Interestingly, comparing Fig. 3 (e) and (f) with Fig. 5 (e) and (f), we see that the pure dephasing time induced by LO phonon vibration increases linearly with increasing dot size even though the exciton-LO-phonon coupling strength decreases exponentially. This result indicates that the dephasing time depends on the exciton-phonon coupling strength nonlinearly. After looking at the phonon induced dephasing time at T ¼ 0K, we now turn to the case at finite temperatures. In Fig. 6, we plot the pure dephasing time of semiconductor QDs induced by the vibration of LO phonon as a function of both dot size and temperature. From Fig. 6, we see that the phonon induced decoherence rate increases with increasing temperature as the results of phonon number increasing. Moreover, the temperature dependence of pure dephasing time of large QDs is stronger than their corresponding small ones. We explain the size dependent temperature behaviors of pure dephasing time to the nonlinear relation between coupling strength and dephasing time. With increasing temperature, the effects of excitonphonon coupling increases at the same rate for QDs of all sizes. Since the contributions on the decoherence process of lattice vibration do not linearly increase with exciton-phonon coupling strength, the dephasing time of large QDs (with relatively weak coupling strength) decreases more rapidly than the small ones (with relatively strong coupling strength) with increasing temperature. This results in a strong temperature dependence of dephasing time in large QDs.
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
57
Fig. 4. Ultrafast dephasing process of the excitonic states of GaP QD with radium of 4 nm induced by LO phonon vibrations.
Fig. 5. Pure dephasing time of IIIeV and IIeVI QDs induced by LA phonon at (a) and (b) X-point, (c) and (d) L-point, and (e) and (f) by LO phonon around the centre of the BZ.
4. Summary In summary, we study the pure dephasing process of spherical colloidal semiconductor QDs using continuum model calculations along with the time-dependent Liouville-von Neumann equation. Based on our calculations, we obtain the
58
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
Fig. 6. Pure dephasing time of IIIeV and IIeVI QDs induced by LO phonon as a function of QD size and temperature.
ultrafast Rabi oscillation of the off-diagonal term r12 between the excitonic state and the virtual state via absorption/emission a phonon and extract the pure dephasing time from the envelope of the oscillations. Our model calculation results show that the vibrations of LO phonon mode dominates the pure dephasing processes of QDs we studied, and the pure dephasing time increases nonlinearly with the decrease of exciton-phonon coupling strength as QD size increasing. We further study the temperature effects on the pure dephasing process of semiconductor QDs and find the temperature dependence of dephasing time is stronger in larger QDs. Acknowledgements This work was funded by the 973 Program of China (No. 2013CBA01702); National Natural Science Foundation of China (No. 11404224, 61205097, 11374216, 91233202, and 11474206); General program of science and technology development project of Beijing Municipal Education Commission under Grant No. KM201510028004; National High Technology Research and Development Program of China (No. 2012AA101608-6); Beijing Natural Science Foundation (No. 1132011); Program for New Century Excellent Talents in University (NCET-12-0607); Program for Beijing Excellent Talents (No. 2013D005016000008); and Scientific Research Base Development Program of the Beijing Municipal Commission of Education. References [1] [2] [3] [4]
C. Burda, X.B. Chen, R. Narayanan, M.A. EI-Sayed, Chem. Rev. 105 (2005) 1025. D.V. Talapin, J.-S. Lee, M.V. Kovalenko, E.V. Shevchenko, Chem. Rev. 110 (2010) 389. re, L. Li, Small 5 (2009) 154. P. Reiss, M. Protie J. Tang, K.W. Kemp, S. Hoogland, K.S. Jeong, H. Liu, L. Levina, M. Furukawa, X. Wang, R. Debnath, D. Cha, K.W. Chou, A. Fischer, A. Amassian, J.B. Asbury, E.H. Sargent, Nat. Mater 10 (2011) 765.
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
Y. Shirasaki, G.J. Supran, M.G. Bawendi, V. Bulovi c, Nat. Photonics 7 (2013) 13. E.H. Sargent, Nat. Photonics 6 (2012) 133. I.J. Kramer, E.H. Sargent, Chem. Rev. 114 (2014) 863. J.L. Herek, W. Wohlleben, R.J. Cogdell, D. Zeidler, M. Motzkus, Nature 417 (2002) 533. J. Liu, S.V. Kilina, S. Tertiak, O.V. Prezhdo, ACS Nano 9 (2015) 9106. B. Krummheuer, V.M. Axt, T. Kuhn, Phys. Rev. B 65 (2002) 195313. E.A. Muljarov, R. Zimmermann, Phys. Rev. Lett. 93 (2004) 237401. G. Lindwall, A. Wacker, C. Weber, A. Knorr, Phys. Rev. Lett. 99 (2007) 087401. C. Roy, S. Hughes, Phys. Rev. Lett. 106 (2011) 247403. € fling, D.P.S. McCutcheon, A. Nazir, Phys. Rev. Lett. 113 (2014) 097401. Y.-J. Wei, Y. He, Y.-M. He, C.-Y. Lu, J.-W. Pan, C. Schneider, M. Kamp, S. Ho S. Kilina, D. Kilin, S. Tretiak, Chem. Rev. 115 (2015) 5929. Z. Nie, R. Long, J. Li, Y.Y. Zheng, O.V. Prezhdo, Z.-H. Loh, J. Phys. Chem. Lett. 4 (2013) 4260. A.B. Madrid, K. Hyeon-Deuk, B.F. Habenicht, O.V. Prezhdo, ACS Nano 3 (2009) 2487. S. Adachi, Properties of Group-IV, IIIeV and IIeVI Semiconductors, Wiley, Chichester, 2005. Y. Hinuma, A. Grüneis, G. Kresse, F. Oba, Phys. Rev. B 90 (2014) 155405. J.C. Merle, R. Sooryakumar, M. Cardona, Phys. Rev. B 30 (1984) 3261. B.H. Lee, J. Appl. Phys. 41 (1970) 2984. P. Han, G. Bester, Phys. Rev. B 83 (2011) 174304. C.G. Van de Walle, Phys. Rev. B 39 (1989) 1871. D. Berlincourt, H. Jaffe, L.R. Shiozawa, Phys. Rev. 129 (1963) 1009. G.D. Mahan, Many-Particle Physics, Kluwer, New York, 2000, pp. 32e36. H. Bruus, K. Flensberg, H. Smith, Phys. Rev. B 48 (1993) 11144. J. Jiang, R. Saito, K. Sato, J.S. Park, Ge G. Samsonidze, A. Jorio, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 75 (2007) 035405. A.R.P. Rau, W. Zhao, Phys. Rev. A 68 (2003) 052102. ^as, S.E. Ulloa, A.O. Govorov, Phys. Rev. B 75 (2007) 155334. J.M. Villas-Bo ^as, A.O. Govorov, S.E. Ulloa, Phys. Rev. B 69 (2004) 125342. J.M. Villas-Bo J.J. Sakurai, in: Modern Quantum Mechanics, Addison-Wesley, Reading, 1994, p. 96.
59
Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Phonon induced pure dephasing process of excitonic state in colloidal semiconductor quantum dots Tongyun Huang, Peng Han*, Xinke Wang, Shengfei Feng, Wenfeng Sun, Jiasheng Ye, Yan Zhang** Department of Physics, Beijing Key Lab for Metamaterials and Devices and Key Laboratory of Terahertz Optoelectronics Ministry of Education, Capital Normal University, Beijing 100048, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 January 2016 Accepted 29 January 2016 Available online 8 February 2016
We present a theoretical study on the pure dephasing process of colloidal semiconductor quantum dots induced by lattice vibrations using continuum model calculations. By solving the time dependent Liouville-von Neumann equation, we present the ultrafast Rabi oscillations between excitonic state and virtual state via exciton-phonon interaction and obtain the pure dephasing time from the fast decayed envelope of the Rabi oscillations. The interaction between exciton and longitudinal optical phonon vibration is found to dominate the pure dephasing process and the dephasing time increases nonlinearly with the reduction of exciton-phonon coupling strength. We further find that the pure dephasing time of large quantum dots is more sensitive to temperature than small quantum dots. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Pure dephasing Semiconductor nanocrystals Ultrafast process
1. Introduction Research on colloidal semiconductor quantum dots (QDs) or called as nanoclusters (NCs) is a rapidly growing field driven by their size tunable electronic and optical properties for the applications as next generation optoelectronics, spintronics, photovoltaics and biolabeling devices [1e7]. Indeed, these applications are usually performed at room temperature where lattice vibrations (phonons) and their interactions with electronic states are naturally involved. In semiconductor QDs, the excited state can be relaxed to the ground state with energy loss to heat via inelastic exciton-phonon scattering, while the phase of the electronic wavefunction is randomized via elastic exciton-phonon interaction and results in loss of quantum coherence, i.e. pure dephasing. The pure dephasing process of QDs is schematically illustrated as a three-level system in Fig. 1. As shown in this figure, the QD is excited from the ground state jg〉 to the excitonic state je1 〉 by a pump pulse. Shortly after the excitation, the system relaxes to state je2 〉 at the bottom of the energy surface via emission phonons. Due to the large energy gap between the excited state je2 〉 and phonon energy, the energy of je2 〉 state is merely shifted slightly without transfer to the other states via coupling to lattice vibrations. During the interaction between excitonic state je2 〉 and lattice vibrations, the phase of the excitonic state wavefunction is destroyed and the quantum coherence of the excited state is lost. In the real applications of nanoscale optoelectronics, photovoltaics, and biological devices, a long-lived quantum coherence of the excited state is partly important for their role in light-harvesting function [8,9]. Moreover, a long-lived quantum coherence is
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (P. Han), [email protected] (Y. Zhang). http://dx.doi.org/10.1016/j.spmi.2016.01.042 0749-6036/© 2016 Elsevier Ltd. All rights reserved.
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
53
Fig. 1. Schematic of the phonon induced pure dephasing process of the excitonic state.
also necessarily needed for the realization of single-photon sources, entangled photons sources, quantum gates and registers in the area of quantum information science and technology. To investigate the pure dephasing process of semiconductor QDs, considerable attentions have been paid from theoretical aspect since the beginning of this century. Based on the definition of quantum decoherence, the dephasing rate of semiconductor self-assemble QDs was firstly calculated via solving the dipole-dipole auto-correlation function using model calculations [10e12]. Later, master equations were applied to study the ultrafast dephasing process of self-assemble QD systems under the excitation of laser field [13,14]. In addition to the model calculations, ab initio time-domain density functional theory (TDDFT) was recently proposed to study the pure dephasing process of semiconductor nanostructures composed by tens to hundreds of atoms [9,15e17]. Although the pure dephasing processes of semiconductor QDs and their effects on the zero phonon line width, Mollow sideband dephasing, and the change of absorption line shape have been studied in the previous researches, a systematical study on the pure dephasing processes of spherical colloidal QDs via exciton acoustic and optical phonon scattering and as a function of QD sizes and temperature is still lacking. In this work, we study the ultrafast pure dephasing process of GaAs, GaP, InP, CdTe, ZnSe and ZnTe spherical colloidal QDs with radius from 3 to 10 nm induced by exciton-phonon interaction via continuum model calculations. The excitonic states of QDs are obtained from effective mass approximation and the exciton-phonon coupling strength are calculated using the €hlich interaction for optical phonon modes. deformation and the piezoelectric potentials for acoustic phonon modes and Fro By solving the Liouville-von Neumann equation with the calculated exciton-phonon coupling strength, the ultrafast pure dephasing process of QDs is obtained. Based on our calculation, we find that: (i) the longitudinal optical (LO) phonon vibrations dominate the pure dephasing process of QDs we studied; (ii) the pure dephasing time increases nonlinear with the decrease of exciton-phonon coupling strength; and (iii) the pure dephasing time of large QDs is more strongly temperature dependent than that of the small dots. 2. Theoretical methods The excitonic state, which is composed by an interacting electron-hole pair, of semiconductor QD with potential barrier Vb is described as,
"
# Z2 2 Z2 2 e2 J ¼ EJ V V þ Vb 2me e 2mh h 4pε0 εr jre rh j
(1)
where, Z denotes the reduced Planck constant, me and mh are the effective masses of electron and hole, ε0 is the vacuum permittivity, εr is the static dielectric constant, e is the electron charge, r e and r h are the positions of the electron and hole, respectively. By solving Eq. (1), the exciton energy E and the corresponding wavefunction of spherical QDs Jnlm ðrÞ ¼ Rnl ðrÞYlm ðq; 4Þ with Rnl ðrÞ the radial function and Ylm ðq; 4Þ the spherical harmonic function, are calculated. The opt optical band gaps of QDs Egap ¼ Ee þ Eh þ Eg E, with Ee the confinement energy of electron, Eh the confinement energy of hole and Eg the band gap of bulk material, are then obtained. In the case of long wavelength acoustic phonons, the atomic displacements correspond to the deformation of lattice and the energy change of the excitonic state induced by the static lattice distortion can be approximated as [25],
54
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
b DF ¼ H
sffiffiffiffiffiffiffiffiffiffiffiffi Z X y VD q$x aq eiq$r þ aq eiq$r 2rVu q
(2)
where V is the volume of QD, r represents the mass density, u is the acoustic phonon frequency, VD denotes the deformation potential induced by static distortion of lattice, q represents the phonon wave vector, x denotes the phonon polarization unit vector, ayq and aq are the creation and the annihilation operators of the acoustic phonon, respectively. In addition to the deformation potential, a stress can also induce a macroscopic electric polarization field in materials without inversion symmetry. The QDs we studied in this work are composed by semiconductors with zinc-blende structure (Td point group symmetry), where only one piezoelectric constant element is independent, d14 ¼ d25 ¼ d36 . The change of excitonic state energy induced by the piezoelectric potential is then written as [26],
b PZ ¼ i H
sffiffiffiffiffiffiffiffiffiffiffiffi Z X y 2ed14 qx qy xz þ qy qz xx þ qz qx xy aq eiq$r þ aq eiq$r 2rVu q
(3)
Combining the deformation and the piezoelectric potentials, the exciton-acoustic phonon interaction matrix Mexph is calculated as [27],
DF b b PZ J〉: Mexph ¼ 〈J H þH
(4)
In addition to the acoustic phonon modes, the relative displacement of oppositely charged atoms, which is induced by the longitudinal optical (LO) phonon vibration in polar materials, generates a macroscopic electric field. The interaction between €hlich interaction with the form [25], this electric field and exciton is known as the Fro
b LO ¼ H
X q
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pe2 ZuLO 1 1 y iq$r aq e þ aq eiq$r 2 ε∞ ε0 εr ε0 q V
(5)
where uLO is the LO phonon frequency and ε∞ is the high frequency dielectric constant. All these calculations are performed using parameters given in Table 1. Once the exciton-phonon coupling matrix elements are obtained, we solve the ultrafast dephasing process of the excited state je2 〉 using the Liouville-von Neumann equation [28,29], Table 1 Material parameters. Material
GaAs
GaP
InP
CdTe
ZnSe
ZnTe
Effective electron mass m*e ðm0 Þ Effective hole mass (heavy) m*hh ðm0 Þ Effective hole mass (light) m*lh ðm0 Þ Potential barrier Vb ðeV Þ Longitudinal sound velocity vl ð105 cm=sÞ Transverse sound velocity vt ð105 cm=sÞ Mass density rðg=cm3 Þ The frequency of LA phonon at X point uLO ðcm1 Þ The frequency of LA phonon at G point uG ðcm1 Þ The frequency of LO phonon uLO ðcm1 Þ Lattice constant að AÞ Static dielectric constant εr high frequency dielectric constant ε∞ Deformation potential for electron VDe ðeV Þ Deformation potential for hole VDh ðeVÞ Piezoelectric constant d14 ð109 V=mÞ Band gap Eg ðeVÞ
0.063a 0.51a 0.082a 4.10c 4.73a 3.35a 5.32a 213g 210g 285b 5.65b 12.9a 10.9b 7.17h 1.16h 1.40a 1.43b
0.090a 0.79a 0.140a 3.58c 5.83a 4.12a 4.14a 246g 233g 403b 5.45b 11.1a 8.8b 8.20h 1.70h 1.02a 2.76b
0.080a 0.60a 0.089a 4.40c 4.58a 3.08a 4.18a 184g 174g 346b 5.87b 12.5a 9.9b 6.00h 0.60h 0.32a 1.35b
0.090b 0.82b 0.145b 4.23c 3.35d 1.79d 5.86f 123g 108g 167b 6.48b 10.2f 7.1b 3.96h 0.55h 0.39i 1.51b
0.137b 0.82b 0.154b 3.99c 4.04e 2.78e 5.26f 184g 167g 252b 5.67b 9.1f 5.9b 4.17h 1.65h 0.61i 2.72b
0.117b 0.67b 0.159b 3.46c 3.55e 2.36e 5.65f 144g 136g 210b 6.01b 7.4f 6.9b 5.83h 0.79h 0.31i 2.27b
a b c d e f g h i
See website: http://www.ioffe.ru/SVA/NSM/Semicond/. Reference [18]. Reference [19]. Reference [20]. Reference [21]. See website: http://www.semiconductors.co.uk/propiivi5410.htm. Reference [22]. Reference [23]. Reference [24].
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
i vb r ðtÞ i hb b ¼ H; r ðtÞ vt Z
55
(6)
The model Hamiltonian of our three-level system is written as [30],
X
b HðtÞ ¼
j¼g;e1 ;e2
1 ðjev 〉〈e2 j þ je2 〉〈ev jÞ þ ZU eiuL t jg〉〈e1 j þ eiuL t je1 〉〈gj Ej jj〉〈jj þ Mexph Nph þ 2
(7)
where Ej represents the eigenvalues of the j th state, jg〉, je1 〉, je2 〉 and jev 〉 are, respectively, the ground state, the excited state, the excitonic state at the bottom of energy surface and the excited virtual state, uL is the frequency of the pulsed pump laser, and U denotes dipole coupling between the light and the system, Nph ¼ 1=expðZuph =kB TÞ 1 is the number of phonon with vibrational frequency uph at temperature with T and kB is the Boltzmann constant. Using the unitary transformation b ¼ exp½iuL t=2ðje 〉〈e j jg〉〈gj þ je 〉〈e jÞ [30] along with the Baker-Hausdorff lemma [31], the Hamiltonian (7) is further U 1 1 2 2 written as,
0
d1 1B b B H ¼ @ ZU 2 0
2ZU d1 2Mexph Nph þ 1=2
1 0 2Mexph Nph þ 1=2 C C A d2
(8)
with energy detuning d1 ¼ Zðue1 ug uL Þ and d2 ¼ d1 þ Zðue2 ue1 Þ, respectively. By solving Eq. (6) with Hamiltonian given in Eq. (8), the ultrafast time dependent dephasing process of the excited state r12 ðtÞ ¼ je1 〉〈e2 j is obtained. 3. Results and discussion The calculated optical band gaps of semiconductor QDs are plotted in Fig. 2 as a function of dot size. We see from this figure that the optical band gap of QDs we studied decreases with dot size increases from 3 to 10 nm and the minimum optical band gap of the QDs we studied (InP QD with 10 nm radium) is around 1.4 eV. Comparing to the phonon energy of QDs (maximum as 50 meV for LO phonon of GaP QD), the excited state cannot be relaxed to the ground state via exciton-phonon coupling due to the large energy detuning and only the coherence of the exciton states losses. Before looking at the pure dephasing process of QDs, we plot the coupling strength of exciton-phonon interaction induced by the LA phonon at the X- and L-points of the Brillouin zone (BZ) along with that induced by the LO phonon around the BZ centre as a function of dot size in Fig. 3. We see from this figure that the strength of exciton-phonon coupling to optical phonon mode is around two orders stronger than that to acoustic phonon mode for QDs we studied. The large exciton-LO phonon coupling is attributed to the strong macroscopic field induced by the relative displacement of cations and anions in the polar IIIeV and IIeVI QDs. Comparing to the strong coupling between exciton and LO phonons, the coupling strength of exciton and LA phonons is much weak for both X- and L-points and increases more rapidly than that to optical phonon mode with decreasing dot size. Once the exciton-phonon coupling strength is obtained, we calculate the pure dephasing process of the excited state using Liouville-von Neumann equation and plot the oscillation process of r12 ðtÞ of GaP QD with radium as 4 nm induced by LO phonon vibrations with time range from 0 to 200 ps in Fig. 4. We see from this figure that the off-diagonal element r12 decays from 0.5 to 0.24 within 200 ps with an exponential envelope function and the dephasing time of the excited state t ¼ 265 ps. To further study the ultrafast dephasing process of the excited state, we plot oscillation of r12 ðtÞ with time from 0 to 0.12 ps as insert in Fig. 4. From this insert figure, we see an ultrafast oscillation between zero to its maximum value within 20 fs. This
Fig. 2. The calculated optical band gap of (a) IIIeV and (b) IIeVI semiconductor QDs as a function of dot size.
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T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
Fig. 3. The size dependent exciton-phonon coupling strength for LA phonon at (a) X-point, (b) L-point, and for (c) LO phonon around the centre of the BZ.
ultrafast oscillation corresponds to the transition between the excited state je2 〉 and the virtual state and the reduction of the oscillation amplitude reflects the loss of coherence. We further plot the pure dephasing time t of GaAs, GaP, InP, CdTe, ZnSe and ZnTe QDs induced by LA phonon at L- and Xpoint and by LO phonon around BZ centre as a function of dot size in Fig. 5 (a)-(f). From this figure, we see that the pure dephasing time induced by acoustic phonon vibration is around few microseconds while that induced by LO phonon is around hundreds of picoseconds for the QDs we studied. Moreover, the pure dephasing time increases with increasing dot size. Both of these characters are determined by the behaviors of exciton-phonon coupling strength, i.e. strong exciton-LOphonon coupling and decrease of exciton-phonon coupling strength with dot size. Interestingly, comparing Fig. 3 (e) and (f) with Fig. 5 (e) and (f), we see that the pure dephasing time induced by LO phonon vibration increases linearly with increasing dot size even though the exciton-LO-phonon coupling strength decreases exponentially. This result indicates that the dephasing time depends on the exciton-phonon coupling strength nonlinearly. After looking at the phonon induced dephasing time at T ¼ 0K, we now turn to the case at finite temperatures. In Fig. 6, we plot the pure dephasing time of semiconductor QDs induced by the vibration of LO phonon as a function of both dot size and temperature. From Fig. 6, we see that the phonon induced decoherence rate increases with increasing temperature as the results of phonon number increasing. Moreover, the temperature dependence of pure dephasing time of large QDs is stronger than their corresponding small ones. We explain the size dependent temperature behaviors of pure dephasing time to the nonlinear relation between coupling strength and dephasing time. With increasing temperature, the effects of excitonphonon coupling increases at the same rate for QDs of all sizes. Since the contributions on the decoherence process of lattice vibration do not linearly increase with exciton-phonon coupling strength, the dephasing time of large QDs (with relatively weak coupling strength) decreases more rapidly than the small ones (with relatively strong coupling strength) with increasing temperature. This results in a strong temperature dependence of dephasing time in large QDs.
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
57
Fig. 4. Ultrafast dephasing process of the excitonic states of GaP QD with radium of 4 nm induced by LO phonon vibrations.
Fig. 5. Pure dephasing time of IIIeV and IIeVI QDs induced by LA phonon at (a) and (b) X-point, (c) and (d) L-point, and (e) and (f) by LO phonon around the centre of the BZ.
4. Summary In summary, we study the pure dephasing process of spherical colloidal semiconductor QDs using continuum model calculations along with the time-dependent Liouville-von Neumann equation. Based on our calculations, we obtain the
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T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59
Fig. 6. Pure dephasing time of IIIeV and IIeVI QDs induced by LO phonon as a function of QD size and temperature.
ultrafast Rabi oscillation of the off-diagonal term r12 between the excitonic state and the virtual state via absorption/emission a phonon and extract the pure dephasing time from the envelope of the oscillations. Our model calculation results show that the vibrations of LO phonon mode dominates the pure dephasing processes of QDs we studied, and the pure dephasing time increases nonlinearly with the decrease of exciton-phonon coupling strength as QD size increasing. We further study the temperature effects on the pure dephasing process of semiconductor QDs and find the temperature dependence of dephasing time is stronger in larger QDs. Acknowledgements This work was funded by the 973 Program of China (No. 2013CBA01702); National Natural Science Foundation of China (No. 11404224, 61205097, 11374216, 91233202, and 11474206); General program of science and technology development project of Beijing Municipal Education Commission under Grant No. KM201510028004; National High Technology Research and Development Program of China (No. 2012AA101608-6); Beijing Natural Science Foundation (No. 1132011); Program for New Century Excellent Talents in University (NCET-12-0607); Program for Beijing Excellent Talents (No. 2013D005016000008); and Scientific Research Base Development Program of the Beijing Municipal Commission of Education. References [1] [2] [3] [4]
C. Burda, X.B. Chen, R. Narayanan, M.A. EI-Sayed, Chem. Rev. 105 (2005) 1025. D.V. Talapin, J.-S. Lee, M.V. Kovalenko, E.V. Shevchenko, Chem. Rev. 110 (2010) 389. re, L. Li, Small 5 (2009) 154. P. Reiss, M. Protie J. Tang, K.W. Kemp, S. Hoogland, K.S. Jeong, H. Liu, L. Levina, M. Furukawa, X. Wang, R. Debnath, D. Cha, K.W. Chou, A. Fischer, A. Amassian, J.B. Asbury, E.H. Sargent, Nat. Mater 10 (2011) 765.
T. Huang et al. / Superlattices and Microstructures 92 (2016) 52e59 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
Y. Shirasaki, G.J. Supran, M.G. Bawendi, V. Bulovi c, Nat. Photonics 7 (2013) 13. E.H. Sargent, Nat. Photonics 6 (2012) 133. I.J. Kramer, E.H. Sargent, Chem. Rev. 114 (2014) 863. J.L. Herek, W. Wohlleben, R.J. Cogdell, D. Zeidler, M. Motzkus, Nature 417 (2002) 533. J. Liu, S.V. Kilina, S. Tertiak, O.V. Prezhdo, ACS Nano 9 (2015) 9106. B. Krummheuer, V.M. Axt, T. Kuhn, Phys. Rev. B 65 (2002) 195313. E.A. Muljarov, R. Zimmermann, Phys. Rev. Lett. 93 (2004) 237401. G. Lindwall, A. Wacker, C. Weber, A. Knorr, Phys. Rev. Lett. 99 (2007) 087401. C. Roy, S. Hughes, Phys. Rev. Lett. 106 (2011) 247403. € fling, D.P.S. McCutcheon, A. Nazir, Phys. Rev. Lett. 113 (2014) 097401. Y.-J. Wei, Y. He, Y.-M. He, C.-Y. Lu, J.-W. Pan, C. Schneider, M. Kamp, S. Ho S. Kilina, D. Kilin, S. Tretiak, Chem. Rev. 115 (2015) 5929. Z. Nie, R. Long, J. Li, Y.Y. Zheng, O.V. Prezhdo, Z.-H. Loh, J. Phys. Chem. Lett. 4 (2013) 4260. A.B. Madrid, K. Hyeon-Deuk, B.F. Habenicht, O.V. Prezhdo, ACS Nano 3 (2009) 2487. S. Adachi, Properties of Group-IV, IIIeV and IIeVI Semiconductors, Wiley, Chichester, 2005. Y. Hinuma, A. Grüneis, G. Kresse, F. Oba, Phys. Rev. B 90 (2014) 155405. J.C. Merle, R. Sooryakumar, M. Cardona, Phys. Rev. B 30 (1984) 3261. B.H. Lee, J. Appl. Phys. 41 (1970) 2984. P. Han, G. Bester, Phys. Rev. B 83 (2011) 174304. C.G. Van de Walle, Phys. Rev. B 39 (1989) 1871. D. Berlincourt, H. Jaffe, L.R. Shiozawa, Phys. Rev. 129 (1963) 1009. G.D. Mahan, Many-Particle Physics, Kluwer, New York, 2000, pp. 32e36. H. Bruus, K. Flensberg, H. Smith, Phys. Rev. B 48 (1993) 11144. J. Jiang, R. Saito, K. Sato, J.S. Park, Ge G. Samsonidze, A. Jorio, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 75 (2007) 035405. A.R.P. Rau, W. Zhao, Phys. Rev. A 68 (2003) 052102. ^as, S.E. Ulloa, A.O. Govorov, Phys. Rev. B 75 (2007) 155334. J.M. Villas-Bo ^as, A.O. Govorov, S.E. Ulloa, Phys. Rev. B 69 (2004) 125342. J.M. Villas-Bo J.J. Sakurai, in: Modern Quantum Mechanics, Addison-Wesley, Reading, 1994, p. 96.
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