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Anomalous polarization in coupled quantum dots
Physica B 279 (2000) 214}216
Anomalous polarization in coupled quantum dots X.H. Xu!,", H. Jiang!,", X. Sun!,",*, H.Q. Lin# !Research Center for Theoretical Physics and Department of Physics, Fudan University, Shanghai 200433, People's Republic of China "National Laboratory of Infrared Physics, Academia Sinica, Shanghai 200083, People's Republic of China #Department of Physics, Chinese University of Hong Kong, Hong Kong, People's Republic of China
Abstract The coupled quantum dots can be designed to possess negative polarizability in low-lying excited states. In an electric "eld, the coupled dots are polarized, and the dipole moment of the coupled dots is reversed by absorbing one photon. This photoswitch e!ect is a new photoinduced phenomenon. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Quantum dot; Polarization; Photoexcitation
1. Introduction The time resolution of the femtosecond technology has reached 10~15 s, meanwhile the lifetime of the excited state is about a nanosecond (10~9 s), which is a very long period in the scale of femtosecond. Hence, various ultrafast processes in the excited states can be revealed by the femtosecond technology [1,2], and the studies of photoinduced phenomenon have found that the excited states possess many novel properties, which cannot exist in the ground state. This paper shows a novel phenomenon in the excitation of the coupled quantum dots. It is the photoinduced polarization inversion } the dipole moment of the coupled dots is reversed by absorbing a photon. As is well known, in magnetism, the molecules with magnetic moment M have positive susceptibility (paramagnetic), and the molecules without M have negative one (dia-magnetic). But, in electricity, both polar and non-polar molecules have positive polarizability a. So there is no terminology of para and dia in electricity [3]. Recently, Sun and his collaborators showed that, in a low-lying excited state, the polymer with a bipolaron can have negative polarizability [4].
* Corresponding author. Tel.: #86-21-65641718; fax: #8621-65104949. E-mail address: [email protected] (X. Sun)
The quantum systems such as the atom, molecule and quantum dot, etc., possess a series of eigenstates t , each i one having its own polarizability a [5] i a " + 2DP D2/(E !E ), i ij j i jEi
(1)
where E is the energy level of t , and P the dipole i i ij between t and t . From Eq. (1), it is seen that i j the polarizability of the ground state of any system is always positive. For the polarizability a of the excited % state t , the states below (above) t give negative (posit% % ive) contributions to a . Since there are only several states % below t but in"nite states above t , even for the excited % % states, usually their polarizability is still positive. However, Eq. (1) cannot guarantee that the polarizability of any excited state must be positive. Actually, the polarizability of the "rst excited state t in a quantum 1 system is negative if it has such an energy spectrum: t is 1 close to the ground state t , and all other excited states ' are much higher, i.e. E !E is much smaller than other 1 ' energy di!erences E !E . In this case, for the polariza% 1 tion a of t , there is a leading term in the Eq. (1), 1 1 which comes from t , other terms are much smaller, ' then a " + 2DP D2/(E !E ) 1 1j j 1 jE1 +2DP D2/(E !E )(0. 1' ' 1
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 7 4 0 - 1
(2)
X.H. Xu et al. / Physica B 279 (2000) 214}216
Therefore, if we can get a quantum system with the spectrum described above, it is possible to obtain the negative polarizability in a low-lying excited state. The quantum dot has an advantage that its structure is #exible. The quantum dot in semiconductors is an arti"cial atom with an adjustable energy spectrum, and the number of electron in this &atom' can be controlled by the grid voltage [6,7]. For a single dot, the con"nement can make the level separation quite big. When two identical dots are coupled, their degenerate ground states are split into a couple of states with opposite parity. For a weak coupling, these two states are close to each other, but separated far away from the other higher states. In this couple of states, the lower one is the ground state, and the upper one the "rst excited state. According to the above analysis, the upper state possesses negative polarizability. Now put an electron into these coupled dots and apply an electric "eld E on it, the coupled dots are polarized. When the electron stays in the ground state, the polarizability of these coupled dots is positive, and their dipole is in the direction of E. When the electron absorbs a photon and transits to the "rst excited state, due to the negative polarizability in t , the dipole of the coupled dots turns 1 over to the opposite direction of E. This is the photoinduced polarization inversion.
215
Fig. 1. The dependences of the "rst four levels on the barrier height < . 0
2. Model In order to simplify the calculation, a dot can be modeled by a three-dimensional rectangular well. When the sizes in > and Z directions are much smaller than in the X direction, the excitation energies caused by the con"nements in the > and Z directions are very high, the low-lying levels are determined by the potential well in the X direction. When two dots approach each other, they are coupled, the coupling of these two dots can be described by a potential barrier between them, then the coupled dots can be modeled by the following potential:
G
< , 0 <(x)" 0,
DxD(a, a)DxD)2a,
(3)
Fig. 2. The polarizabilities of the ground state and "rst excited state.
two dots couple to each other, two-fold degenerate ground state is split into E and E . It should be men' 1 tioned that, in weak coupling, the energy di!erence between E and E is much smaller than that between other 1 ' levels.
R, DxD'2a.
In this model, the width of each dot is a, the width and height of the barrier are 2a and < , respectively, which 0 control the coupling strength. The energy levels and wave functions in this potential well can be calculated analytically. Fig. 1 shows the dependences of the "rst four levels on < , where E "(p+)2/16mHa2 is the energy of the 0 0 ground state at < "0, mH is the e!ective mass of an 0 electron. In Fig. 1, we can see that, when < is very large, the 0 coupling is very weak, level E is almost equal to E . In 1 ' this case, the two dots are actually independent and their ground states are degenerate. When < decreases, these 0
3. Polarizability From Eq. (1), the polarizability of the state t in the i X direction is a " + 2DS j Dex DiT D2/(E !E ). i j i jEi
(4)
It is straightforward to calculate the dipole moment S jDex DiT from known eigenfunctions. The polarizabilities a and a of the ground state and "rst excited state are ' 1 shown in Fig. 2, where a "8(ea)2/E . 0 0
216
X.H. Xu et al. / Physica B 279 (2000) 214}216
From Fig. 2, it is seen that a is positive but a nega' 1 tive. These results are expected from the analysis in the introduction. When the barrier height < increases, the 0 energy di!erence between E and E decreases, and the 1 ' absolute values of a and a become larger. ' 1
Acknowledgements X. Sun is extremely grateful to the Physics Department of the Chinese University of Hong Kong for hospitality during his visit there. This work was supported by the NSFC Grants (59790050, 19874014), 863-715-10 and Shanghai Center of Applied Physics.
4. Conclusion The negative polarizability can be realized in the "rst excited state of the coupled quantum dots. When one electron is injected into this system by adjusting the grid voltage, and an electric "eld is applied, the coupled dots are polarized and a dipole moment is induced. In the ground state, since its polarizability is positive, the dipole is in the direction of the electric "eld. By using photoexcitation, the electron absorbs one photon to be excited into the "rst excited state. Since the polarizability of the "rst excited state is negative, the dipole points to the opposite direction of the electric "eld. It indicates that the dipole of the coupled dots is reversed by absorbing a photon. It is the photoinduced polarization inversion.
References [1] S. Watanabe, Frontiers in Laser Physics and Spectroscopy, Pergamon, Oxford, 1996. [2] H. Stapelfeldt, E. Constant, P.B. Corkum, Prog. Crystal. Growth 33 (1996) 209. [3] K.D. Bonin, V.V. Kresin, Electric-dipole Polarizabilities of Atoms, Molecules, and Clusters, World Scienti"c, Singapore, 1997. [4] X. Sun et al., Proceedings of Organic Solids Symposium, Okazaki, 1998. [5] L.D. Landau, E.M. Lifshitz, Quantnm Mechanics, Pergamon Press, Oxford, 1977. [6] M.A. Reed et al., Phys. Rev. Lett. 60 (1988) 535. [7] K. Ismail et al., Appl. Phys. Lett. 54 (1989) 460.
Anomalous polarization in coupled quantum dots X.H. Xu!,", H. Jiang!,", X. Sun!,",*, H.Q. Lin# !Research Center for Theoretical Physics and Department of Physics, Fudan University, Shanghai 200433, People's Republic of China "National Laboratory of Infrared Physics, Academia Sinica, Shanghai 200083, People's Republic of China #Department of Physics, Chinese University of Hong Kong, Hong Kong, People's Republic of China
Abstract The coupled quantum dots can be designed to possess negative polarizability in low-lying excited states. In an electric "eld, the coupled dots are polarized, and the dipole moment of the coupled dots is reversed by absorbing one photon. This photoswitch e!ect is a new photoinduced phenomenon. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Quantum dot; Polarization; Photoexcitation
1. Introduction The time resolution of the femtosecond technology has reached 10~15 s, meanwhile the lifetime of the excited state is about a nanosecond (10~9 s), which is a very long period in the scale of femtosecond. Hence, various ultrafast processes in the excited states can be revealed by the femtosecond technology [1,2], and the studies of photoinduced phenomenon have found that the excited states possess many novel properties, which cannot exist in the ground state. This paper shows a novel phenomenon in the excitation of the coupled quantum dots. It is the photoinduced polarization inversion } the dipole moment of the coupled dots is reversed by absorbing a photon. As is well known, in magnetism, the molecules with magnetic moment M have positive susceptibility (paramagnetic), and the molecules without M have negative one (dia-magnetic). But, in electricity, both polar and non-polar molecules have positive polarizability a. So there is no terminology of para and dia in electricity [3]. Recently, Sun and his collaborators showed that, in a low-lying excited state, the polymer with a bipolaron can have negative polarizability [4].
* Corresponding author. Tel.: #86-21-65641718; fax: #8621-65104949. E-mail address: [email protected] (X. Sun)
The quantum systems such as the atom, molecule and quantum dot, etc., possess a series of eigenstates t , each i one having its own polarizability a [5] i a " + 2DP D2/(E !E ), i ij j i jEi
(1)
where E is the energy level of t , and P the dipole i i ij between t and t . From Eq. (1), it is seen that i j the polarizability of the ground state of any system is always positive. For the polarizability a of the excited % state t , the states below (above) t give negative (posit% % ive) contributions to a . Since there are only several states % below t but in"nite states above t , even for the excited % % states, usually their polarizability is still positive. However, Eq. (1) cannot guarantee that the polarizability of any excited state must be positive. Actually, the polarizability of the "rst excited state t in a quantum 1 system is negative if it has such an energy spectrum: t is 1 close to the ground state t , and all other excited states ' are much higher, i.e. E !E is much smaller than other 1 ' energy di!erences E !E . In this case, for the polariza% 1 tion a of t , there is a leading term in the Eq. (1), 1 1 which comes from t , other terms are much smaller, ' then a " + 2DP D2/(E !E ) 1 1j j 1 jE1 +2DP D2/(E !E )(0. 1' ' 1
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 7 4 0 - 1
(2)
X.H. Xu et al. / Physica B 279 (2000) 214}216
Therefore, if we can get a quantum system with the spectrum described above, it is possible to obtain the negative polarizability in a low-lying excited state. The quantum dot has an advantage that its structure is #exible. The quantum dot in semiconductors is an arti"cial atom with an adjustable energy spectrum, and the number of electron in this &atom' can be controlled by the grid voltage [6,7]. For a single dot, the con"nement can make the level separation quite big. When two identical dots are coupled, their degenerate ground states are split into a couple of states with opposite parity. For a weak coupling, these two states are close to each other, but separated far away from the other higher states. In this couple of states, the lower one is the ground state, and the upper one the "rst excited state. According to the above analysis, the upper state possesses negative polarizability. Now put an electron into these coupled dots and apply an electric "eld E on it, the coupled dots are polarized. When the electron stays in the ground state, the polarizability of these coupled dots is positive, and their dipole is in the direction of E. When the electron absorbs a photon and transits to the "rst excited state, due to the negative polarizability in t , the dipole of the coupled dots turns 1 over to the opposite direction of E. This is the photoinduced polarization inversion.
215
Fig. 1. The dependences of the "rst four levels on the barrier height < . 0
2. Model In order to simplify the calculation, a dot can be modeled by a three-dimensional rectangular well. When the sizes in > and Z directions are much smaller than in the X direction, the excitation energies caused by the con"nements in the > and Z directions are very high, the low-lying levels are determined by the potential well in the X direction. When two dots approach each other, they are coupled, the coupling of these two dots can be described by a potential barrier between them, then the coupled dots can be modeled by the following potential:
G
< , 0 <(x)" 0,
DxD(a, a)DxD)2a,
(3)
Fig. 2. The polarizabilities of the ground state and "rst excited state.
two dots couple to each other, two-fold degenerate ground state is split into E and E . It should be men' 1 tioned that, in weak coupling, the energy di!erence between E and E is much smaller than that between other 1 ' levels.
R, DxD'2a.
In this model, the width of each dot is a, the width and height of the barrier are 2a and < , respectively, which 0 control the coupling strength. The energy levels and wave functions in this potential well can be calculated analytically. Fig. 1 shows the dependences of the "rst four levels on < , where E "(p+)2/16mHa2 is the energy of the 0 0 ground state at < "0, mH is the e!ective mass of an 0 electron. In Fig. 1, we can see that, when < is very large, the 0 coupling is very weak, level E is almost equal to E . In 1 ' this case, the two dots are actually independent and their ground states are degenerate. When < decreases, these 0
3. Polarizability From Eq. (1), the polarizability of the state t in the i X direction is a " + 2DS j Dex DiT D2/(E !E ). i j i jEi
(4)
It is straightforward to calculate the dipole moment S jDex DiT from known eigenfunctions. The polarizabilities a and a of the ground state and "rst excited state are ' 1 shown in Fig. 2, where a "8(ea)2/E . 0 0
216
X.H. Xu et al. / Physica B 279 (2000) 214}216
From Fig. 2, it is seen that a is positive but a nega' 1 tive. These results are expected from the analysis in the introduction. When the barrier height < increases, the 0 energy di!erence between E and E decreases, and the 1 ' absolute values of a and a become larger. ' 1
Acknowledgements X. Sun is extremely grateful to the Physics Department of the Chinese University of Hong Kong for hospitality during his visit there. This work was supported by the NSFC Grants (59790050, 19874014), 863-715-10 and Shanghai Center of Applied Physics.
4. Conclusion The negative polarizability can be realized in the "rst excited state of the coupled quantum dots. When one electron is injected into this system by adjusting the grid voltage, and an electric "eld is applied, the coupled dots are polarized and a dipole moment is induced. In the ground state, since its polarizability is positive, the dipole is in the direction of the electric "eld. By using photoexcitation, the electron absorbs one photon to be excited into the "rst excited state. Since the polarizability of the "rst excited state is negative, the dipole points to the opposite direction of the electric "eld. It indicates that the dipole of the coupled dots is reversed by absorbing a photon. It is the photoinduced polarization inversion.
References [1] S. Watanabe, Frontiers in Laser Physics and Spectroscopy, Pergamon, Oxford, 1996. [2] H. Stapelfeldt, E. Constant, P.B. Corkum, Prog. Crystal. Growth 33 (1996) 209. [3] K.D. Bonin, V.V. Kresin, Electric-dipole Polarizabilities of Atoms, Molecules, and Clusters, World Scienti"c, Singapore, 1997. [4] X. Sun et al., Proceedings of Organic Solids Symposium, Okazaki, 1998. [5] L.D. Landau, E.M. Lifshitz, Quantnm Mechanics, Pergamon Press, Oxford, 1977. [6] M.A. Reed et al., Phys. Rev. Lett. 60 (1988) 535. [7] K. Ismail et al., Appl. Phys. Lett. 54 (1989) 460.