- Email: [email protected]
Influence of temperature on the excited state of the strong coupling polaron in a CsI quantum pseudodot
Accepted Manuscript
Influence of temperature on the excited state of the strong coupling polaron in a CsI quantum pseudodot Jian-Fang Zhang , Jing-Lin Xiao PII: DOI: Reference:
S0577-9073(16)30348-3 10.1016/j.cjph.2016.07.012 CJPH 84
To appear in:
Chinese Journal of Physics
Received date: Accepted date:
2 July 2016 25 July 2016
Please cite this article as: Jian-Fang Zhang , Jing-Lin Xiao , Influence of temperature on the excited state of the strong coupling polaron in a CsI quantum pseudodot, Chinese Journal of Physics (2016), doi: 10.1016/j.cjph.2016.07.012
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights Polaron effect in CsI quantum pseudodot (QPD) with pseudoharmonic potential. Using variational method of Pekar type, first-excited state energy was studied. Excitation energy of strong-coupling polaron in the CsI QPD. Transition frequency of strong-coupling polaron in the CsI QPD Using a quantum statistical theory, polaron temperature effect was calculated.
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Influence of temperature on the excited state of the strong coupling polaron in a CsI quantum pseudodot Jian-Fang Zhang and Jing-Lin Xiao*
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Institute of Condensed Matter Physics, Inner Mongolia University for the Nationalities, Tongliao, 028043, China *[email protected] Abstract In the present work, we have considered a polaron confined in a CsI quantum
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pseudodot (QPD) with the pseudoharmonic potential (PHP). The properties of the first-excited state energy, the excitation energy and the transition frequency of the strong-coupling polaron in the CsI QPD were studied by using variational method of Pekar type (VMPT). By employing a quantum statistical theory (QST), we have calculated the effect of the temperature on the properties of the polaron. The investigated results demonstrate that the absolute value of first-excited state energy, the excitation energy and the transition frequency lift (decay) with raising temperature in lower (higher) temperature regime. The excitation energy and the transition frequency are enhancing functions of the chemical potential of the two-dimensional electron gas, whereas the absolute value of first-excited state energy is a decaying one of it.
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Keywords CsI Quantum pseudodot Temperature Quantum statistical theory Excited state PACS numbers: 71.38.-k, 73.21.la, 63.20.kd
1 Introduction
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With the great development of the nanoscience and nanotechnology, the studies of low dimensional nanostructures have become an interesting research field in the recently decades. These nanostructures (or quantum structures), for instance superlattices, quantum wires, quantum rods and quantum dots (QDs) [1–4], have transition energy levels and particular electronic properties, so the polaron effects in these structures are getting stronger and more important. A large number of experiments and the theoretical works of those systems are performed for expounding the physics of the systems. In the first place of experiments, using far-infrared magnetospectroscopy, Hameau et al. [5] investigated the electronic transitions from the ground s levels to the excited p levels. The experiments consist of monitoring, by means of Zeeman tuning of the excited level, a resonant interaction between the discrete (p, 0 LO phonon) state and the continuum of either (s, 1 LO phonon) or (s, 2 LO phonons). They also showed that the electrons and the LO phonons are always in a strong coupling regime and form an everlasting mixed electron-phonon mode. Thus, they found evidence for the everlasting resonant polarons of strong electron-phonon coupling regime in quantum dots. Bissiri et al. [6] performed photoluminescence (PL) and resonant PL at low temperatures in a number of InAs/GaAs QDs whose emission energies range from 1.4 to 1.08 eV, found that the value of the electron-phonon interaction S is large for small QDs and small values for large well-formed QDs, and showed the trend is consistent with recent experimental results in InAs QDs and provides an experimental basis to recent theoretical speculations. Thus, they found that the optical evidence of polaron interaction in 2
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InAs/GaAs QDs. Preisler et al. [7] investigated the interband transitions in several ensembles of self-assembled InAs/GaAs QDs by using photoluminescence excitation spectroscopy under strong magnetic field, determined the excitonic polaron energies as well as the oscillator strengths of the interband transitions and found the good agreement between the calculations and the experimental data evidencing the existence of excitonic polarons InAs/GaAs QDs. From these examples, we found that the research of polaron effects in different crystal materials (such as InAs, GaAs, and others) nanostructures is of prime importance. Just in the same way of the theories, using the compact density-matrix approach and the iterative method, Li et al. [8] theoretically investigated the polaron effect on the optical rectification in a two-dimensional QPD system under the influence of a staticmagnetic field. Khordad [9] used the Lee–Low–Pines unitary transformation method and the Pekar type variational procedure to study the influence of the Rashba effect on bound polaron in a QPD. Khordad et al. [10] studied the influence of electron–phonon (e–p) interaction on third-harmonic generation in a GaAs QPD. From the theories, we found that the polaron effect and electron-phonon interaction in the different material QPD is interesting for researching low dimensional nanostructures. Recently, Ma et al. [11] used a variational method of the Pekar type to investigate coulomb impurity potential RbCl QPD qubit. Xiao [12] employed the same method to study the effect of magnetic field on RbCl QPD qubit. However, the research of the temperature effect on the excited state of the strong coupling polaron in a CsI QPD is seldom. In the present work, the properties of the first-excited state energy, the excitation energy and the transition frequency of the strong-coupling polaron in the CsI QPD were studied by using VMPT. Considering polaron confined in CsI QPDs with the PHP and employing a quantum statistical theory (QST), we have calculated the effect of the temperature on these properties of the polaron. The results are useful and important for the nanoscience and nanotechnology, especially meaning for low dimensional nanostructures field.
2 Theoretical models
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We consider an electron, which is interacting with bulk LO phonons and subject to a pseudoharmonic potential (PHP). The Hamiltonian of the electron-phonon system in the CsI QPDs has the following forms:
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p2 H V r LO aq aq Vq aq exp iq r h.c , 2m q q
(1)
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where
2
r r V r V0 0 , r0 r
(2)
where the significances of the physical quantities in Eq.(1) are the same with Ref.[13]. V r is the PHP [14] that includes both harmonic QD potential and antidot potential. V0 and r0 are the chemical potential of the two-dimensional electron gas and the zero point of the PHP, respectively. Following the variational method of Pekar type ( VMPT) [15-17] and the Ref. [18], the trial wavefunction of the strong-coupling polaron can be separated into two parts, which individually 3
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describes the electron and the phonon, the trial wavefunction can be expressed as
= U 0 ph ,
depends only on the electron coordinate, 0 ph
aq 0 ph 0 , and U 0 ph
is the phonon’s vacuum state with
denotes the coherent state of the phonon,
U exp aq f q aq f q* , q
(4)
*
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where
(3)
where f q ( f q ) is the variational function, we choose the trial polaron’s wavefunction of the ground and the first excited states (GFES) as [19]
3
3
02 r 2 0 ph , 2
1 1 1 0 ph where
1
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0 0 0 0 ph 4 02 exp
2r 2 3 4 52 1 r cos exp 1 exp i 0 ph , 2 4
(5)
(6)
0 and 1 are the variational parameters. By the VMPT, the polaron’s energies of the
first excited state and the excitation energy in the CsI QPD can be calculated using
E E1 E0
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5V 5 2 2 2 3 2 E1 1 1 20 2 V012 r02 2V0 LO 1R0 , 4m 21 r1 3 4
3 2 2 LO 1 R0 LO 0 R0 , 4
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(8)
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5V 3V 5 2 2 3 2 2 2 1 0 20 2 20 2 V012 r02 2V002 r02 4m 4m 21 r1 20 r0 3
(7)
where R0
2mLO
1
2
is the polaron radius. The transition frequency of the polaron reads
AC
as follows:
E
.
(9)
The mean number of LO phonons around the electron can be described as
N 0 U 1 aq aqU 0 q
1 3 R00 R01. 2 4 2
(10)
3 Temperature effects At finite temperature, the lattice vibrations excite not only the real phonon but also electron in a 4
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PHP. The statistical average value of different states can be used to describe the properties of polaron. According to the QST, statistical average number of the bulk LO phonons can be obtained as
N exp LO K BT 1
1
(11)
where K B and T present the Boltzmann constant and the temperature of the system. Through self-consistently calculation based on Eq. (10) and Eq. (11), we can obtain the relationship
0 and 1 and the temperature T . From Eqs. (7), (8) and (9)
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between variational parameters
we can see that the ground state energy, the first excited state energy and the mean number of bulk LO phonons of strong-coupling polaron in a QPD with PHP depends on the variational parameters 0 and 1 , and then is related to the temperature T .
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4 Numerical results and discussion
In this section, we carry out numerical calculations for a CsI QPD crystal. The parameters used in our calculations are
LO 11.22845266meV , m 0.42m0 and 3.67 [20].
Figure 1 plots the first-excited state energy E1 as a functions of the temperature T and the
M
chemical potential of the two-dimensional electron gas V0 for R0 10.0nm and r 0 12.0nm .
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Figs. 2 and 3 indicate respectively the relationship between the excitation energy E and the transition frequency of the polaron varying with the temperature T and the chemical
PT
potential of the two-dimensional electron gas V0 for R0 8.0nm and r 0 12.0nm . From Figs. 1, 2 and 3, we can see that the absolute value of first-excited state energy E1 , the
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excitation energy E and the transition frequency are an increasing (a decreasing ) functions of the temperature T at lower ( higher ) temperature regime. This is due to with the lift of the temperature, the velocities of the electrons and the phonons are raised, and then there will be more phonons to interact with electrons. While in lower temperature regime, the contribution from the increased electron velocity that makes more electrons to be located on the first-excited state, is larger than that from the electrons interacting with more phonons, which destroys the first-excited state. So the absolute value of first-excited state energy, the excitation energy and the transition frequency are enhanced. However, the contributions from the electron velocity and the electron-phonon interaction are reversed, leading to the decrease of they in higher temperature region. From them we can also see that the excitation energy and the transition frequency are an enhancing function of the chemical potential of the two-dimensional electron gas, whereas the absolute value of first-excited state energy is a decaying one of it. With the increasing of the chemical potential of the two-dimensional electron gas, the PHP is enhanced, and then the energy 5
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of the electron and the interaction between the electron and the phonon are raised. From another point of view, since the presence of the PHP is equivalent to introduce another confinement on the electron, which leads to greater electron wavefunction overlapping with each other, the energy of the electron and the electron-phonon interactions will be amplified, and the excitation energy and the transition frequency are an enhancing functions of the chemical potential of the two-dimensional electron gas. However the second, the three and the four terms in the Eq. (7) are the contribution of the chemical potential of the two-dimensional electron gas to the first-excited state energy, which are a positive value, whereas the total first-excited state energy is negative value. Wherefore, the absolute value of first-excited state energy is a decaying function of it. . As suggested above, we can adjust the first-excited state energy, the excitation energy and the transition frequency of the strong-coupling polaron in the CsI QPD by changing the temperature and the chemical potential of the two-dimensional electron gas. In summary, the temperature, the pseudoharmonic potential and the chemical potential of the two-dimensional electron gas are important factors to study the properties of the polaron in a CsI QPD with hydrogen-like impurity.
5 Conclusions
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By using VMPT, we calculated the relationship between the first-excited state energy, the excitation energy and the transition frequency of the strong-coupling polaron in the CsI QPD varying with the chemical potential of the two-dimensional electron gas. The influences of the temperature on the first-excited state energy, the excitation energy and the transition frequency were investigated by using QST. We find that ① the absolute value of first-excited state energy, the excitation energy and the transition frequency are an increasing (a decreasing ) functions of the temperature at lower ( higher ) temperature regime. ② the excitation energy and the transition frequency are an enhancing functions of the chemical potential of the two-dimensional electron gas, whereas the absolute value of first-excited state energy is a decaying one of it.
Acknowledgements
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This project was supported by the National Science Foundation of China under Grant No.11464033.
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References
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[1] N. Vukmirović, Z. Ikonić, D. Indjin, et al., Phys. Rev. B 76, 245313 (2007). [2] A. L. Vartanian, M. A. Yeranosyan, A. A. Kirakosyan, Physica. E 27, 447 (2005). [3] J. L. Xiao and Z. H. Ding, J. Low Temp. Phys. 163, 302 (2011). [4] M. A. Odnoblyudov, I. N. Yassievich, and K. A. Chao, Phys. Rev. Lett. 83, 4884 (1999). [5] S. Hameau, Y. Guldner, O. Verzelen, et al. Phys. Rev. Lett. 83, 4152 (1999). [6] M. Bissiri, G. B. H. von Högersthal, A. S. Bhatti, et al. Phys. Rev. B 62, 4642 (2000). [7] V. Preisler, T. Grange, R. Ferreira, et al. Phys. Rev. B 73, 075320 (2006). [8] N. Li, K. X. Guo, and S. Shao, Opt. Quant. Electron. 44, 493 (2012). [9] R. Khordad, Physica E 69, 249 (2015). [10] R. Khordad, B. Mirhosseini, and H. Bahramiyan, Opt. Quant. Electron. 48, 1 (2016). [11] J. L. Xiao, Mod. Phys. Lett. B 29, 1550098 (2015). [12] X. J. Ma, B. Qi, and J. L. Xiao, J. Low Temp. Phys. 180, 315 (2015). [13] Y. Sun, Z. H. Ding, and J. L. Xiao, J. Electron. Mater. 45, 3576 (2016). [14] A. Cetin, Phys. Lett. A 372, 3852 (2008) [15] L.D. Landau and S.I. Pekar, Zh. Eksp. Teor. Fiz.18, 419 (1948) [16] S. I. Pekar and M. F. Deigen, Zh. Eksp. Teor. Fiz. 18, 481(1948) [17] S. I. Pekar, Untersuchungen über die Elektronen-theorie der Kristalle, Akademie Verlag, Berlin (1954) [18] W. Xiao, B. Qi, and J. L. Xiao, J. Low. Temp. Phys. 179, 166 (2015) [19] Z. H. Ding, Y. Sun, and J. L. Xiao, Int. J. Quantum. Inf. 10, 1250077 (2012) [20] J.T. Devreese, Polarons in ionic crystals and polar semiconductors, North-Holland, Amsterdam (1972)
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Fig.1 The first-excited state energy E1 versus the temperature T and the chemical
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potential of the two-dimensional electron gas V0 .
Fig.2 The excitation energy E versus the temperature T and the chemical potential of the two-dimensional electron gas V0 ..
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Fig.3 The transition frequency versus the temperature T and
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the chemical potential of the two-dimensional electron gas V0 .
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Influence of temperature on the excited state of the strong coupling polaron in a CsI quantum pseudodot Jian-Fang Zhang , Jing-Lin Xiao PII: DOI: Reference:
S0577-9073(16)30348-3 10.1016/j.cjph.2016.07.012 CJPH 84
To appear in:
Chinese Journal of Physics
Received date: Accepted date:
2 July 2016 25 July 2016
Please cite this article as: Jian-Fang Zhang , Jing-Lin Xiao , Influence of temperature on the excited state of the strong coupling polaron in a CsI quantum pseudodot, Chinese Journal of Physics (2016), doi: 10.1016/j.cjph.2016.07.012
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Highlights Polaron effect in CsI quantum pseudodot (QPD) with pseudoharmonic potential. Using variational method of Pekar type, first-excited state energy was studied. Excitation energy of strong-coupling polaron in the CsI QPD. Transition frequency of strong-coupling polaron in the CsI QPD Using a quantum statistical theory, polaron temperature effect was calculated.
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Influence of temperature on the excited state of the strong coupling polaron in a CsI quantum pseudodot Jian-Fang Zhang and Jing-Lin Xiao*
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Institute of Condensed Matter Physics, Inner Mongolia University for the Nationalities, Tongliao, 028043, China *[email protected] Abstract In the present work, we have considered a polaron confined in a CsI quantum
AN US
pseudodot (QPD) with the pseudoharmonic potential (PHP). The properties of the first-excited state energy, the excitation energy and the transition frequency of the strong-coupling polaron in the CsI QPD were studied by using variational method of Pekar type (VMPT). By employing a quantum statistical theory (QST), we have calculated the effect of the temperature on the properties of the polaron. The investigated results demonstrate that the absolute value of first-excited state energy, the excitation energy and the transition frequency lift (decay) with raising temperature in lower (higher) temperature regime. The excitation energy and the transition frequency are enhancing functions of the chemical potential of the two-dimensional electron gas, whereas the absolute value of first-excited state energy is a decaying one of it.
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Keywords CsI Quantum pseudodot Temperature Quantum statistical theory Excited state PACS numbers: 71.38.-k, 73.21.la, 63.20.kd
1 Introduction
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With the great development of the nanoscience and nanotechnology, the studies of low dimensional nanostructures have become an interesting research field in the recently decades. These nanostructures (or quantum structures), for instance superlattices, quantum wires, quantum rods and quantum dots (QDs) [1–4], have transition energy levels and particular electronic properties, so the polaron effects in these structures are getting stronger and more important. A large number of experiments and the theoretical works of those systems are performed for expounding the physics of the systems. In the first place of experiments, using far-infrared magnetospectroscopy, Hameau et al. [5] investigated the electronic transitions from the ground s levels to the excited p levels. The experiments consist of monitoring, by means of Zeeman tuning of the excited level, a resonant interaction between the discrete (p, 0 LO phonon) state and the continuum of either (s, 1 LO phonon) or (s, 2 LO phonons). They also showed that the electrons and the LO phonons are always in a strong coupling regime and form an everlasting mixed electron-phonon mode. Thus, they found evidence for the everlasting resonant polarons of strong electron-phonon coupling regime in quantum dots. Bissiri et al. [6] performed photoluminescence (PL) and resonant PL at low temperatures in a number of InAs/GaAs QDs whose emission energies range from 1.4 to 1.08 eV, found that the value of the electron-phonon interaction S is large for small QDs and small values for large well-formed QDs, and showed the trend is consistent with recent experimental results in InAs QDs and provides an experimental basis to recent theoretical speculations. Thus, they found that the optical evidence of polaron interaction in 2
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InAs/GaAs QDs. Preisler et al. [7] investigated the interband transitions in several ensembles of self-assembled InAs/GaAs QDs by using photoluminescence excitation spectroscopy under strong magnetic field, determined the excitonic polaron energies as well as the oscillator strengths of the interband transitions and found the good agreement between the calculations and the experimental data evidencing the existence of excitonic polarons InAs/GaAs QDs. From these examples, we found that the research of polaron effects in different crystal materials (such as InAs, GaAs, and others) nanostructures is of prime importance. Just in the same way of the theories, using the compact density-matrix approach and the iterative method, Li et al. [8] theoretically investigated the polaron effect on the optical rectification in a two-dimensional QPD system under the influence of a staticmagnetic field. Khordad [9] used the Lee–Low–Pines unitary transformation method and the Pekar type variational procedure to study the influence of the Rashba effect on bound polaron in a QPD. Khordad et al. [10] studied the influence of electron–phonon (e–p) interaction on third-harmonic generation in a GaAs QPD. From the theories, we found that the polaron effect and electron-phonon interaction in the different material QPD is interesting for researching low dimensional nanostructures. Recently, Ma et al. [11] used a variational method of the Pekar type to investigate coulomb impurity potential RbCl QPD qubit. Xiao [12] employed the same method to study the effect of magnetic field on RbCl QPD qubit. However, the research of the temperature effect on the excited state of the strong coupling polaron in a CsI QPD is seldom. In the present work, the properties of the first-excited state energy, the excitation energy and the transition frequency of the strong-coupling polaron in the CsI QPD were studied by using VMPT. Considering polaron confined in CsI QPDs with the PHP and employing a quantum statistical theory (QST), we have calculated the effect of the temperature on these properties of the polaron. The results are useful and important for the nanoscience and nanotechnology, especially meaning for low dimensional nanostructures field.
2 Theoretical models
PT
We consider an electron, which is interacting with bulk LO phonons and subject to a pseudoharmonic potential (PHP). The Hamiltonian of the electron-phonon system in the CsI QPDs has the following forms:
CE
p2 H V r LO aq aq Vq aq exp iq r h.c , 2m q q
(1)
AC
where
2
r r V r V0 0 , r0 r
(2)
where the significances of the physical quantities in Eq.(1) are the same with Ref.[13]. V r is the PHP [14] that includes both harmonic QD potential and antidot potential. V0 and r0 are the chemical potential of the two-dimensional electron gas and the zero point of the PHP, respectively. Following the variational method of Pekar type ( VMPT) [15-17] and the Ref. [18], the trial wavefunction of the strong-coupling polaron can be separated into two parts, which individually 3
ACCEPTED MANUSCRIPT
describes the electron and the phonon, the trial wavefunction can be expressed as
= U 0 ph ,
depends only on the electron coordinate, 0 ph
aq 0 ph 0 , and U 0 ph
is the phonon’s vacuum state with
denotes the coherent state of the phonon,
U exp aq f q aq f q* , q
(4)
*
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where
(3)
where f q ( f q ) is the variational function, we choose the trial polaron’s wavefunction of the ground and the first excited states (GFES) as [19]
3
3
02 r 2 0 ph , 2
1 1 1 0 ph where
1
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0 0 0 0 ph 4 02 exp
2r 2 3 4 52 1 r cos exp 1 exp i 0 ph , 2 4
(5)
(6)
0 and 1 are the variational parameters. By the VMPT, the polaron’s energies of the
first excited state and the excitation energy in the CsI QPD can be calculated using
E E1 E0
ED
M
5V 5 2 2 2 3 2 E1 1 1 20 2 V012 r02 2V0 LO 1R0 , 4m 21 r1 3 4
3 2 2 LO 1 R0 LO 0 R0 , 4
CE
(8)
PT
5V 3V 5 2 2 3 2 2 2 1 0 20 2 20 2 V012 r02 2V002 r02 4m 4m 21 r1 20 r0 3
(7)
where R0
2mLO
1
2
is the polaron radius. The transition frequency of the polaron reads
AC
as follows:
E
.
(9)
The mean number of LO phonons around the electron can be described as
N 0 U 1 aq aqU 0 q
1 3 R00 R01. 2 4 2
(10)
3 Temperature effects At finite temperature, the lattice vibrations excite not only the real phonon but also electron in a 4
ACCEPTED MANUSCRIPT
PHP. The statistical average value of different states can be used to describe the properties of polaron. According to the QST, statistical average number of the bulk LO phonons can be obtained as
N exp LO K BT 1
1
(11)
where K B and T present the Boltzmann constant and the temperature of the system. Through self-consistently calculation based on Eq. (10) and Eq. (11), we can obtain the relationship
0 and 1 and the temperature T . From Eqs. (7), (8) and (9)
CR IP T
between variational parameters
we can see that the ground state energy, the first excited state energy and the mean number of bulk LO phonons of strong-coupling polaron in a QPD with PHP depends on the variational parameters 0 and 1 , and then is related to the temperature T .
AN US
4 Numerical results and discussion
In this section, we carry out numerical calculations for a CsI QPD crystal. The parameters used in our calculations are
LO 11.22845266meV , m 0.42m0 and 3.67 [20].
Figure 1 plots the first-excited state energy E1 as a functions of the temperature T and the
M
chemical potential of the two-dimensional electron gas V0 for R0 10.0nm and r 0 12.0nm .
ED
Figs. 2 and 3 indicate respectively the relationship between the excitation energy E and the transition frequency of the polaron varying with the temperature T and the chemical
PT
potential of the two-dimensional electron gas V0 for R0 8.0nm and r 0 12.0nm . From Figs. 1, 2 and 3, we can see that the absolute value of first-excited state energy E1 , the
AC
CE
excitation energy E and the transition frequency are an increasing (a decreasing ) functions of the temperature T at lower ( higher ) temperature regime. This is due to with the lift of the temperature, the velocities of the electrons and the phonons are raised, and then there will be more phonons to interact with electrons. While in lower temperature regime, the contribution from the increased electron velocity that makes more electrons to be located on the first-excited state, is larger than that from the electrons interacting with more phonons, which destroys the first-excited state. So the absolute value of first-excited state energy, the excitation energy and the transition frequency are enhanced. However, the contributions from the electron velocity and the electron-phonon interaction are reversed, leading to the decrease of they in higher temperature region. From them we can also see that the excitation energy and the transition frequency are an enhancing function of the chemical potential of the two-dimensional electron gas, whereas the absolute value of first-excited state energy is a decaying one of it. With the increasing of the chemical potential of the two-dimensional electron gas, the PHP is enhanced, and then the energy 5
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of the electron and the interaction between the electron and the phonon are raised. From another point of view, since the presence of the PHP is equivalent to introduce another confinement on the electron, which leads to greater electron wavefunction overlapping with each other, the energy of the electron and the electron-phonon interactions will be amplified, and the excitation energy and the transition frequency are an enhancing functions of the chemical potential of the two-dimensional electron gas. However the second, the three and the four terms in the Eq. (7) are the contribution of the chemical potential of the two-dimensional electron gas to the first-excited state energy, which are a positive value, whereas the total first-excited state energy is negative value. Wherefore, the absolute value of first-excited state energy is a decaying function of it. . As suggested above, we can adjust the first-excited state energy, the excitation energy and the transition frequency of the strong-coupling polaron in the CsI QPD by changing the temperature and the chemical potential of the two-dimensional electron gas. In summary, the temperature, the pseudoharmonic potential and the chemical potential of the two-dimensional electron gas are important factors to study the properties of the polaron in a CsI QPD with hydrogen-like impurity.
5 Conclusions
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By using VMPT, we calculated the relationship between the first-excited state energy, the excitation energy and the transition frequency of the strong-coupling polaron in the CsI QPD varying with the chemical potential of the two-dimensional electron gas. The influences of the temperature on the first-excited state energy, the excitation energy and the transition frequency were investigated by using QST. We find that ① the absolute value of first-excited state energy, the excitation energy and the transition frequency are an increasing (a decreasing ) functions of the temperature at lower ( higher ) temperature regime. ② the excitation energy and the transition frequency are an enhancing functions of the chemical potential of the two-dimensional electron gas, whereas the absolute value of first-excited state energy is a decaying one of it.
Acknowledgements
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This project was supported by the National Science Foundation of China under Grant No.11464033.
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References
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[1] N. Vukmirović, Z. Ikonić, D. Indjin, et al., Phys. Rev. B 76, 245313 (2007). [2] A. L. Vartanian, M. A. Yeranosyan, A. A. Kirakosyan, Physica. E 27, 447 (2005). [3] J. L. Xiao and Z. H. Ding, J. Low Temp. Phys. 163, 302 (2011). [4] M. A. Odnoblyudov, I. N. Yassievich, and K. A. Chao, Phys. Rev. Lett. 83, 4884 (1999). [5] S. Hameau, Y. Guldner, O. Verzelen, et al. Phys. Rev. Lett. 83, 4152 (1999). [6] M. Bissiri, G. B. H. von Högersthal, A. S. Bhatti, et al. Phys. Rev. B 62, 4642 (2000). [7] V. Preisler, T. Grange, R. Ferreira, et al. Phys. Rev. B 73, 075320 (2006). [8] N. Li, K. X. Guo, and S. Shao, Opt. Quant. Electron. 44, 493 (2012). [9] R. Khordad, Physica E 69, 249 (2015). [10] R. Khordad, B. Mirhosseini, and H. Bahramiyan, Opt. Quant. Electron. 48, 1 (2016). [11] J. L. Xiao, Mod. Phys. Lett. B 29, 1550098 (2015). [12] X. J. Ma, B. Qi, and J. L. Xiao, J. Low Temp. Phys. 180, 315 (2015). [13] Y. Sun, Z. H. Ding, and J. L. Xiao, J. Electron. Mater. 45, 3576 (2016). [14] A. Cetin, Phys. Lett. A 372, 3852 (2008) [15] L.D. Landau and S.I. Pekar, Zh. Eksp. Teor. Fiz.18, 419 (1948) [16] S. I. Pekar and M. F. Deigen, Zh. Eksp. Teor. Fiz. 18, 481(1948) [17] S. I. Pekar, Untersuchungen über die Elektronen-theorie der Kristalle, Akademie Verlag, Berlin (1954) [18] W. Xiao, B. Qi, and J. L. Xiao, J. Low. Temp. Phys. 179, 166 (2015) [19] Z. H. Ding, Y. Sun, and J. L. Xiao, Int. J. Quantum. Inf. 10, 1250077 (2012) [20] J.T. Devreese, Polarons in ionic crystals and polar semiconductors, North-Holland, Amsterdam (1972)
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Fig.1 The first-excited state energy E1 versus the temperature T and the chemical
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potential of the two-dimensional electron gas V0 .
Fig.2 The excitation energy E versus the temperature T and the chemical potential of the two-dimensional electron gas V0 ..
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Fig.3 The transition frequency versus the temperature T and
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the chemical potential of the two-dimensional electron gas V0 .
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