- Email: [email protected]
Polaron couplings in quantum dot molecules
Physica E 13 (2002) 309 – 312
www.elsevier.com/locate/physe
Polaron couplings in quantum dot molecules O. Verzelen ∗ , R. Ferreira, G. Bastard LPMC ENS, 24 rue Lhomond F75005 Paris, France
Abstract We report on the calculation of polaron energies in InAs quantum dot molecules. Polaron e-ects are larger in vertical than in lateral molecules. The far-infrared absorption associated with molecular polaron transitions is calculated. It may show prominent lines associated with inter-dot polaron transitions. We have also calculated the polaron relaxation time to thermodynamical equilibrium when its lifetime is limited by the decay of its phonon component due to crystal anharmonicity. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 71.38.+i; 73.61.+r Keywords: Polaron; Quantum dot; Relaxation; Electron–phonon interaction
1. Introduction
2. Polaron eects in vertical dot molecules
Both experimental and theoretical 9ndings [1–3] have recently pointed out that electrons and longitudinal optical (LO) phonons enter a strong coupling regime in semiconductor quantum dots. This means that the most accurate description of the interacting electrons and LO phonons is not in terms of slightly damped products of electrons and LO phonons states but of coherent admixtures of electron and phonon states: the polarons. Attention was so far focussed on intra-dot e-ects where the electron–phonon coupling a-ects only electron states which belong to a given dot. In this paper, we report on the evaluation of the polaron couplings between the electron states which belong to a dot molecule formed by two di-erent, vertically displaced, dots.
Let us consider two dots separated by a distance d along the growth (z) axis (see inset Fig. 1). The dot radii (heights) are Ru and R‘ (hu and h‘ ), respectively. Each dot is modelled by a truncated cone Aoating on a one monolayer thick (0:333 nm) wetting layer, as approximately realized in the InAs=GaAs self-assembled quantum dots structures. In the following, we shall assume that the dot molecules contain one electron. The con9ning potential is −Ve in the dots and wetting layer, and zero in the GaAs barrier. The barrier edge is taken as the energy zero of the calculations and we take Ve = 0:4 eV and a carrier e-ective mass of 0:07m0 in the following. The LO phonons will be taken bulk-like and dispersionless: ˜!0 = 36 meV. This approximation provides an excellent description of the single-dot polarons [1,2]. Fig. 1 shows the tunnel-coupled electron states with zero phonon (solid lines) and one LO phonon (dashed lines), respectively, but no electron–phonon coupling for two dots separated by d = 10 nm. The lower dot
∗
Corresponding author. Fax: +33-(0)-1-44-32-38-40. E-mail address: [email protected] (O. Verzelen).
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 5 4 5 - 8
310
O. Verzelen et al. / Physica E 13 (2002) 309 – 312 -20
-20
-30
-30
|2l 〉
E(meV)
E(meV)
|S l , 1q 〉 -40
-40
|1l 〉 |2u 〉
|S u , 1q 〉
u -50
-50
d
|1u 〉
l -60 100
102
104
106
108
110
-60 100
102
Ru (Å)
Fig. 1. Solid lines: tunnel-coupled electron states in a vertical dot molecule plotted versus Ru . Dashed lines: tunnel-coupled one phonon levels |Su ; 1q and |S‘ ; 1q . R‘ = 10 nm; d = 10 nm; h =R = 0:13.
has 9xed dimensions (R‘ = 10 nm; h‘ = 1:3 nm) and we vary the dimensions of the upper dot, keeping its aspect ratio constant (hu =Ru = 0:13). The electron states of the molecule have been expanded on a basis spanned by the single dot |S , |Px and |Py states (suitably Schmidt orthogonalized), where = u; ‘. The one-phonon molecule spectrum (dashed lines) is obtained by a rigid shift of the zero-phonon spectrum (solid lines) by ˜!LO . The zero-phonon levels (solid lines) |Pu and |P‘ anticross weakly (≈ 3 meV) each other at Ru =10 nm like the one-phonon levels (dashed lines) |Su ; 1q and |S‘ ; 1q (anticrossing ≈ 2 meV). The electron–phonon interaction is taken to be of the FrGohlich type. The problem one faces is to diagonalize this Hamiltonian on a basis which comprises six discrete levels (the zero-phonon molecule states) and six di-erent three-dimensional (3D) dispersionless continuums (the one-phonon molecule states). For detail, see Ref. [4]. The result of such a diagonalization is shown in Fig. 3 for two dots with the same material parameters as in Fig. 1. For clarity, we show only the polaron levels which contain at least 5% of P component with zero phonon. These are the levels which will be electric dipole-coupled to the ground level to produce the far-infrared absorption of electronic origin. There are many anticrossings together with shifts. A relatively simple picture of the molecular polaron is obtained by building 9rst the one-dot polarons and then let them hybridize by tunnel and inter-dot FrGohlich couplings. In a single dot, the |P
104
106
108
110
R u (Å)
Fig. 2. The Ru dependence of the polaron levels of the isolated dots. Same parameters as in Fig. 1.
levels are coupled with the one-phonon continuum |S; 1. They give rise to two polarons labelled |1 and |2. The other couplings are far from resonance. So for interpretation, we can neglect them. Now, when two polaron levels pertaining to di-erent dots come to cross versus Ru , the inter-dot (tunnel and FrGohlich) couplings lead to anticrossings (unless symmetry forbids it). Fig. 2 displays the Ru dependencies of the polaron levels of the isolated dots. One 9nds the bonding (|1u and |1‘ ) and antibonding (|2u and |2‘ ) polaron states which contain both “atomic” S and P components. In dashed lines we have also drawn the two one-phonon levels of Fig. 1 resulting from the tunnel coupling of |Su ; 1q and |S‘ ; 1q . The two |S1u ; |S1‘ levels have simple variations since they display near Ru = 10 nm the same kind of anticrossing as the two |Su ; 1q and |S‘ ; 1q onephonon levels in Fig. 2. A more interesting situation arises when we consider the mixing between the levels which were |2u , |1‘ ; |Su ; 1q and |S‘ ; 1q . The successive anticrossings give rise to the intricate level scheme of Fig. 3. In order to probe these polaron levels the farinfrared absorption is a convenient tool. For low-temperature experiments we can assume that only the ground state |S0 ≈ |S0u is occupied. The light absorption probability is found proportional to P(!) = |S0 |x||2 L [˜! − (S0 ) − ()]; (1)
where | are the dot molecule excited polaron levels and L (x) is the normalized Lorentzian function with
O. Verzelen et al. / Physica E 13 (2002) 309 – 312
show up. Near 36 meV there is a strong intra-dot peak |S0u → |1u . It is followed by a doublet which is associated with |S0u → | ≈ a |2u + b |1‘ , where a and b take two di-erent values for the two branches ( = ±1) of the anticrossing between |2u and |1‘ . Finally, the small high-energy component corresponds to the transition |S0u → | where | is a mixture between |1‘ , |2u and the continuum states |S‘ ; 1q .
-20
E(meV)
-30
-40
-50
-60 100
102
104
106
108
110
R u (Å)
Fig. 3. Energies of the polaron levels (which contain at least 5% of zero-phonon P components) in a vertical dot molecule plotted versus Ru . Same parameters as in Fig. 1.
Absorption (UA)
12
(a) 8 4 0
30
35
40 Energy (meV)
45
50
Absorption (UA)
9
(b) 6
| σ = +1 〉
3 0
30
311
35
40 Energy (meV)
| σ = _1〉
45
50
Fig. 4. Calculated absorption spectra associated with polaron transitions in a vertical dot molecule. R‘ = 10 nm; d = 10 nm; h =R = 0:13; Ru = 11 nm (a), and Ru = 10:4 nm (b).
full-width at half-maximum 2. In practice, we took = 0:05 meV. Fig. 4a shows the calculated far-infrared spectrum in the case of a dot molecule without any particular anticrossings (Ru = 11 nm in Fig. 3). There are two peaks corresponding, respectively, to the |S0u → |1u and |S0u → |2u transitions, i.e. a typical single-dot situation. The two lines have di-erent oscillator strengths due to a non-resonant intra-dot polaron e-ect. Fig. 4b instead corresponds to an anomalous spectrum (Ru = 10:4 nm in Fig. 3) where four peaks
3. Relaxation and thermalization of polaron states Besides the energetics of the molecular polarons, there are informations one should obtain about the coupling between the polaron molecules and their environment. We have already shown [5] that “atomic” polarons, formally free of relaxation due to the LO phonon emission (since they are the eigenstates of the interacting electron–phonon system), can, in fact, relax to their (polaron) ground state because of the coupling to the phonon thermostat. This coupling originates from the crystal anharmonicity [6,7] and a-ects the polaron through its LO phonon component. This coupling should also be present in molecular polarons. We have investigated the time evolution of a molecular polaron system by means of simpli9ed rate equations in the following manner: The ground state |g contains mostly a zero-phonon component. The excited states |e contains both zero- and one-phonon sizeable components. The anharmonicity couples |e and |g via the surrender to the thermostat of two phonons, one zone edge transverse acoustical and one zone edge optical phonons (like in bulk materials) provided the two levels have an energy di-erence which is in the energy window [28 meV, 45 meV] which allows this process. If so, the population of the upper state is taken to decrease at a rate 1=pol which is taken equal to 1=bulk , the bulk LO phonon 1 rate ( 10 ps at low temperature), times | |2 the square modulus of the one-phonon part of the |e state. Such a semi-classical approach was already used by Li and Arakawa [8] for “atomic” polarons and it was checked [4] that this semi-classical approach supplies relaxation rates which are in reasonable agreement with more microscopic estimates provided the polaron energies di-erence is in the right energy window. Note that each excited polaron level is twice degenerate (on
312
O. Verzelen et al. / Physica E 13 (2002) 309 – 312 1
Population
T = 10K
(a)
0.8
|S l 〉
0.6 0.4
0
|S u 〉
| σ = + 1x 〉
0.2 0
200
400
600
upper level and |S‘ is much more eNcient than that to the ground state |Su . Note, however, that a Boltzmann distribution will be recovered at extremely long time. Upon increasing T , the expected tendency to a Boltzmann-like equilibrium takes place (Fig. 5b) at moderate time.
Time(ps) 1
(b)
Population
0.8
| σ = + 1x 〉
0.6 0.4
|S u 〉
0.2
|S l 〉
0
0
Acknowledgements
T = 200K
200
400
600
Time(ps)
Fig. 5. Calculated time evolution of the populations of polarons in a vertical dot molecule. (a) T = 10 K, and (b) T = 200 K. Same dot parameters as in Fig. 4b.
account of the electronic Px and Py degeneracy). At t = 0 we suppose that all the polaron levels are empty but one (containing e.g. Px and no Py ) | = +1x . The time evolution of the populations taking into account the coupling to the thermostat is presented in Figs. 5a and b, corresponding, respectively, to T = 10 and 200 K. At low temperature (Fig. 5a) and for time in excess of 1 ns, one 9nds a pseudo-equilibrium statistical distribution which is opposite to the Boltzmann expectation. This is because the coupling between the
The LPMC–ENS is a “UnitQe AssociQee au CNRS (UMR 8551) et aux UniversitQes Paris 6 et Paris 7”. This work has been partly supported by a New Energy and Industrial Technology Development Organization (NEDO) Grant and by an E.E.C. Project (IST-1999-11311 (SQID)). We gratefully acknowledge Prs. Y. Guldner, H. Sakaki and Dr. S. Hameau for their useful discussions. References [1] [2] [3] [4] [5]
S. Hameau, et al., Phys. Rev. Lett. 83 (1999) 4152. O. Verzelen, et al., Jpn. J. Appl. Phys. 40 (2001) 1941. T. Inoshita, H. Sakaki, Phys. Rev. B 56 (1997) R4355. O. Verzelen, et al., Phys. Rev. B 64 (2001) 075315. O. Verzelen, R. Ferreira, G. Bastard, Phys. Rev. B 62 (2000) R4809. [6] F. VallQee, F. Bogani, Phys. Rev. B 43 (1991) 12 049. [7] F. VallQee, Phys. Rev. B 49 (1994) 2460. [8] X.Q. Li, Y. Arakawa, Phys. Rev. B 57 (1998) 12 285.
www.elsevier.com/locate/physe
Polaron couplings in quantum dot molecules O. Verzelen ∗ , R. Ferreira, G. Bastard LPMC ENS, 24 rue Lhomond F75005 Paris, France
Abstract We report on the calculation of polaron energies in InAs quantum dot molecules. Polaron e-ects are larger in vertical than in lateral molecules. The far-infrared absorption associated with molecular polaron transitions is calculated. It may show prominent lines associated with inter-dot polaron transitions. We have also calculated the polaron relaxation time to thermodynamical equilibrium when its lifetime is limited by the decay of its phonon component due to crystal anharmonicity. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 71.38.+i; 73.61.+r Keywords: Polaron; Quantum dot; Relaxation; Electron–phonon interaction
1. Introduction
2. Polaron eects in vertical dot molecules
Both experimental and theoretical 9ndings [1–3] have recently pointed out that electrons and longitudinal optical (LO) phonons enter a strong coupling regime in semiconductor quantum dots. This means that the most accurate description of the interacting electrons and LO phonons is not in terms of slightly damped products of electrons and LO phonons states but of coherent admixtures of electron and phonon states: the polarons. Attention was so far focussed on intra-dot e-ects where the electron–phonon coupling a-ects only electron states which belong to a given dot. In this paper, we report on the evaluation of the polaron couplings between the electron states which belong to a dot molecule formed by two di-erent, vertically displaced, dots.
Let us consider two dots separated by a distance d along the growth (z) axis (see inset Fig. 1). The dot radii (heights) are Ru and R‘ (hu and h‘ ), respectively. Each dot is modelled by a truncated cone Aoating on a one monolayer thick (0:333 nm) wetting layer, as approximately realized in the InAs=GaAs self-assembled quantum dots structures. In the following, we shall assume that the dot molecules contain one electron. The con9ning potential is −Ve in the dots and wetting layer, and zero in the GaAs barrier. The barrier edge is taken as the energy zero of the calculations and we take Ve = 0:4 eV and a carrier e-ective mass of 0:07m0 in the following. The LO phonons will be taken bulk-like and dispersionless: ˜!0 = 36 meV. This approximation provides an excellent description of the single-dot polarons [1,2]. Fig. 1 shows the tunnel-coupled electron states with zero phonon (solid lines) and one LO phonon (dashed lines), respectively, but no electron–phonon coupling for two dots separated by d = 10 nm. The lower dot
∗
Corresponding author. Fax: +33-(0)-1-44-32-38-40. E-mail address: [email protected] (O. Verzelen).
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 1 ) 0 0 5 4 5 - 8
310
O. Verzelen et al. / Physica E 13 (2002) 309 – 312 -20
-20
-30
-30
|2l 〉
E(meV)
E(meV)
|S l , 1q 〉 -40
-40
|1l 〉 |2u 〉
|S u , 1q 〉
u -50
-50
d
|1u 〉
l -60 100
102
104
106
108
110
-60 100
102
Ru (Å)
Fig. 1. Solid lines: tunnel-coupled electron states in a vertical dot molecule plotted versus Ru . Dashed lines: tunnel-coupled one phonon levels |Su ; 1q and |S‘ ; 1q . R‘ = 10 nm; d = 10 nm; h =R = 0:13.
has 9xed dimensions (R‘ = 10 nm; h‘ = 1:3 nm) and we vary the dimensions of the upper dot, keeping its aspect ratio constant (hu =Ru = 0:13). The electron states of the molecule have been expanded on a basis spanned by the single dot |S , |Px and |Py states (suitably Schmidt orthogonalized), where = u; ‘. The one-phonon molecule spectrum (dashed lines) is obtained by a rigid shift of the zero-phonon spectrum (solid lines) by ˜!LO . The zero-phonon levels (solid lines) |Pu and |P‘ anticross weakly (≈ 3 meV) each other at Ru =10 nm like the one-phonon levels (dashed lines) |Su ; 1q and |S‘ ; 1q (anticrossing ≈ 2 meV). The electron–phonon interaction is taken to be of the FrGohlich type. The problem one faces is to diagonalize this Hamiltonian on a basis which comprises six discrete levels (the zero-phonon molecule states) and six di-erent three-dimensional (3D) dispersionless continuums (the one-phonon molecule states). For detail, see Ref. [4]. The result of such a diagonalization is shown in Fig. 3 for two dots with the same material parameters as in Fig. 1. For clarity, we show only the polaron levels which contain at least 5% of P component with zero phonon. These are the levels which will be electric dipole-coupled to the ground level to produce the far-infrared absorption of electronic origin. There are many anticrossings together with shifts. A relatively simple picture of the molecular polaron is obtained by building 9rst the one-dot polarons and then let them hybridize by tunnel and inter-dot FrGohlich couplings. In a single dot, the |P
104
106
108
110
R u (Å)
Fig. 2. The Ru dependence of the polaron levels of the isolated dots. Same parameters as in Fig. 1.
levels are coupled with the one-phonon continuum |S; 1. They give rise to two polarons labelled |1 and |2. The other couplings are far from resonance. So for interpretation, we can neglect them. Now, when two polaron levels pertaining to di-erent dots come to cross versus Ru , the inter-dot (tunnel and FrGohlich) couplings lead to anticrossings (unless symmetry forbids it). Fig. 2 displays the Ru dependencies of the polaron levels of the isolated dots. One 9nds the bonding (|1u and |1‘ ) and antibonding (|2u and |2‘ ) polaron states which contain both “atomic” S and P components. In dashed lines we have also drawn the two one-phonon levels of Fig. 1 resulting from the tunnel coupling of |Su ; 1q and |S‘ ; 1q . The two |S1u ; |S1‘ levels have simple variations since they display near Ru = 10 nm the same kind of anticrossing as the two |Su ; 1q and |S‘ ; 1q onephonon levels in Fig. 2. A more interesting situation arises when we consider the mixing between the levels which were |2u , |1‘ ; |Su ; 1q and |S‘ ; 1q . The successive anticrossings give rise to the intricate level scheme of Fig. 3. In order to probe these polaron levels the farinfrared absorption is a convenient tool. For low-temperature experiments we can assume that only the ground state |S0 ≈ |S0u is occupied. The light absorption probability is found proportional to P(!) = |S0 |x||2 L [˜! − (S0 ) − ()]; (1)
where | are the dot molecule excited polaron levels and L (x) is the normalized Lorentzian function with
O. Verzelen et al. / Physica E 13 (2002) 309 – 312
show up. Near 36 meV there is a strong intra-dot peak |S0u → |1u . It is followed by a doublet which is associated with |S0u → | ≈ a |2u + b |1‘ , where a and b take two di-erent values for the two branches ( = ±1) of the anticrossing between |2u and |1‘ . Finally, the small high-energy component corresponds to the transition |S0u → | where | is a mixture between |1‘ , |2u and the continuum states |S‘ ; 1q .
-20
E(meV)
-30
-40
-50
-60 100
102
104
106
108
110
R u (Å)
Fig. 3. Energies of the polaron levels (which contain at least 5% of zero-phonon P components) in a vertical dot molecule plotted versus Ru . Same parameters as in Fig. 1.
Absorption (UA)
12
(a) 8 4 0
30
35
40 Energy (meV)
45
50
Absorption (UA)
9
(b) 6
| σ = +1 〉
3 0
30
311
35
40 Energy (meV)
| σ = _1〉
45
50
Fig. 4. Calculated absorption spectra associated with polaron transitions in a vertical dot molecule. R‘ = 10 nm; d = 10 nm; h =R = 0:13; Ru = 11 nm (a), and Ru = 10:4 nm (b).
full-width at half-maximum 2. In practice, we took = 0:05 meV. Fig. 4a shows the calculated far-infrared spectrum in the case of a dot molecule without any particular anticrossings (Ru = 11 nm in Fig. 3). There are two peaks corresponding, respectively, to the |S0u → |1u and |S0u → |2u transitions, i.e. a typical single-dot situation. The two lines have di-erent oscillator strengths due to a non-resonant intra-dot polaron e-ect. Fig. 4b instead corresponds to an anomalous spectrum (Ru = 10:4 nm in Fig. 3) where four peaks
3. Relaxation and thermalization of polaron states Besides the energetics of the molecular polarons, there are informations one should obtain about the coupling between the polaron molecules and their environment. We have already shown [5] that “atomic” polarons, formally free of relaxation due to the LO phonon emission (since they are the eigenstates of the interacting electron–phonon system), can, in fact, relax to their (polaron) ground state because of the coupling to the phonon thermostat. This coupling originates from the crystal anharmonicity [6,7] and a-ects the polaron through its LO phonon component. This coupling should also be present in molecular polarons. We have investigated the time evolution of a molecular polaron system by means of simpli9ed rate equations in the following manner: The ground state |g contains mostly a zero-phonon component. The excited states |e contains both zero- and one-phonon sizeable components. The anharmonicity couples |e and |g via the surrender to the thermostat of two phonons, one zone edge transverse acoustical and one zone edge optical phonons (like in bulk materials) provided the two levels have an energy di-erence which is in the energy window [28 meV, 45 meV] which allows this process. If so, the population of the upper state is taken to decrease at a rate 1=pol which is taken equal to 1=bulk , the bulk LO phonon 1 rate ( 10 ps at low temperature), times | |2 the square modulus of the one-phonon part of the |e state. Such a semi-classical approach was already used by Li and Arakawa [8] for “atomic” polarons and it was checked [4] that this semi-classical approach supplies relaxation rates which are in reasonable agreement with more microscopic estimates provided the polaron energies di-erence is in the right energy window. Note that each excited polaron level is twice degenerate (on
312
O. Verzelen et al. / Physica E 13 (2002) 309 – 312 1
Population
T = 10K
(a)
0.8
|S l 〉
0.6 0.4
0
|S u 〉
| σ = + 1x 〉
0.2 0
200
400
600
upper level and |S‘ is much more eNcient than that to the ground state |Su . Note, however, that a Boltzmann distribution will be recovered at extremely long time. Upon increasing T , the expected tendency to a Boltzmann-like equilibrium takes place (Fig. 5b) at moderate time.
Time(ps) 1
(b)
Population
0.8
| σ = + 1x 〉
0.6 0.4
|S u 〉
0.2
|S l 〉
0
0
Acknowledgements
T = 200K
200
400
600
Time(ps)
Fig. 5. Calculated time evolution of the populations of polarons in a vertical dot molecule. (a) T = 10 K, and (b) T = 200 K. Same dot parameters as in Fig. 4b.
account of the electronic Px and Py degeneracy). At t = 0 we suppose that all the polaron levels are empty but one (containing e.g. Px and no Py ) | = +1x . The time evolution of the populations taking into account the coupling to the thermostat is presented in Figs. 5a and b, corresponding, respectively, to T = 10 and 200 K. At low temperature (Fig. 5a) and for time in excess of 1 ns, one 9nds a pseudo-equilibrium statistical distribution which is opposite to the Boltzmann expectation. This is because the coupling between the
The LPMC–ENS is a “UnitQe AssociQee au CNRS (UMR 8551) et aux UniversitQes Paris 6 et Paris 7”. This work has been partly supported by a New Energy and Industrial Technology Development Organization (NEDO) Grant and by an E.E.C. Project (IST-1999-11311 (SQID)). We gratefully acknowledge Prs. Y. Guldner, H. Sakaki and Dr. S. Hameau for their useful discussions. References [1] [2] [3] [4] [5]
S. Hameau, et al., Phys. Rev. Lett. 83 (1999) 4152. O. Verzelen, et al., Jpn. J. Appl. Phys. 40 (2001) 1941. T. Inoshita, H. Sakaki, Phys. Rev. B 56 (1997) R4355. O. Verzelen, et al., Phys. Rev. B 64 (2001) 075315. O. Verzelen, R. Ferreira, G. Bastard, Phys. Rev. B 62 (2000) R4809. [6] F. VallQee, F. Bogani, Phys. Rev. B 43 (1991) 12 049. [7] F. VallQee, Phys. Rev. B 49 (1994) 2460. [8] X.Q. Li, Y. Arakawa, Phys. Rev. B 57 (1998) 12 285.