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0 1 2 -8 -6 -4 -2 0 2 4 6 3

1e-08 1e-06 0.0001 0.01 1

energy

αe in ¯hωLO time inps

|GRα(t)|

(a) conduction band

0 1 2 3 -0.6 -0.4

-0.2 0

0.2 0.4 0.0001

0.01 1

energy h1

α inhω¯ LO time inps

|GRα(t)|

(b) upper valence band (VB1)

0 1 2 3

0 1

2 3

0.0001 0.01 1

energy h2

α in¯hωLO time inps

|GRα(t)|

(c) lower valence band (VB2)

Fig. 7.12: Modulus of the polaron retarded GF for the conduction band as well as for the upper and lower valence band.

In Fig. 7.12 the modulus of the retarded GF in the time domain, following from a numer-ical solution of the corresponding Dyson equation, is depicted as a function of time and energy, both for QD and WL states. The energy dispersion taken into account is depicted in Fig. 7.10, whereas other material parameters are summarized in Tab. B.1. For the dis-cussion note, that GaN is an intermediate polar coupling material, α = 0.5, and the LO phonon energy amounts to ωLO = 90 meV.2 As discussed in detail in Chap. 5.1.1, the modulus of the retarded GF reflects the quasi-particle lifetime, whereas its phase describes the quasi-particle energy. Remarkably, despite the large level spacing of approximately 2.55 ωLO for the conduction-band QD states, we find a considerable decay that corre-sponds to a short quasi-particle lifetime. The oscillatory character points towards a

side-2cf. the discussion on the phonon dispersion for nitrides in Chap. B.1.

band structure in the spectral function, which is discussed below. For the conduction-band (CB) WL states a phonon threshold is observed around1ωLO. Below the threshold the decay is significantly slower due to reduced phonon emission processes. For the lower valence-band (VB2) WL states a similar behavior is observed. In contrast, for the upper valence band (VB1) WL states no phonon threshold is present, as the energy dispersion is rather flat. To observe a phonon threshold for a flat dispersion large momentum transfer is necessary, but the corresponding matrix elements are negligible. Note, that especially for higher WL states the calculation contains numerical artifacts as a finer discretization of the WL continuum exceeds our computational capacities.

1e-08 1e-06 0.0001 0.01

1 CB k=0

1e-08 1e-06 0.0001 0.01

1 p-shell

1e-08 1e-06 0.0001 0.01 1

-10 -8 -6 -4 -2 0 2

s-shell

energy in¯LO spectralfunction!Gα(ω)

(a) conduction band

1e-08 1e-06 0.0001 0.01

1 VB1 k=0

VB2 k=0

1e-08 1e-06 0.0001 0.01

1 p-shell

1e-08 1e-06 0.0001 0.01 1

-4 -2 0 2 4

s-shell

energy in¯LO spectralfunction!Gα(ω)

(b) valence band

Fig. 7.13: Spectral function for QD and WLk = 0states of the conduction band and the valence band.

The corresponding spectral functions are presented in Fig. 7.13. For the CB QD states a broad structure, containing multiple phonon satellites as well as hybridization effects (cf.

the discussion in Chap. 5.1.2), is observed. Even though the level spacing is approximately 2.55ωLOfor CB QD states, the QD states strongly interact via multiple-phonon processes, whereas the interaction between QD and WL states (level spacing exceeds 3.8ωLO) is

considerably smaller. In the CB WL spectral function we observe a main peak exhibiting a polaron shift and a broad satellite 1 ωLO energetically above. For the VB states the picture is more involved due to the presence of several subbands. Whereas the VB2 WL spectral function resembles the effects discussed for the CB, the VB1 WL spectral function as well as the VB QD states show a qualitatively different behavior. In contrast to the CB we observe mainly satellites spaced 1 ωLO apart. Since the same physics as in the CB case, i.e. diagonal and off-diagonal coupling of QD states, is involved, one would expect hybridization effects. Note, that also in the spectral function numerical artifacts due to the coarse discretization of the WL continuum are observed.

0.001 0.01 0.1 1 10

-4 -3 -2 -1 0 1 2 3 4 s-shell p-shell

energy in¯hωLO spectralfunction!Gα(ω)

(a) s-p coupling

0.001 0.01 0.1 1 10

-4 -3 -2 -1 0 1 2 3 4 s-shell

k=0

energy in¯hωLO spectralfunction!Gα(ω)

(b) s-WL coupling

0.001 0.01 0.1 1 10

-4 -3 -2 -1 0 1 2 3 4 p-shell

k=0

energy in¯hωLO spectralfunction!Gα(ω)

(c) p-WL coupling

0.01 0.1 1 10 100

-4 -3 -2 -1 0 1 2 3 4 s-shell p-shell

energy in¯hωLO spectralfunction!Gα(ω)

(d) total QD

Fig. 7.14: Contributions to the model spectral function of QD states that interact with a dispersionless WL.

To analyze the situation, we consider a two-level model that can be diagonalized numer-ically, cf. Chap. 5.1.3, where a phenomenological broadening was chosen to resemble the broadening in Fig. 7.13b. The two levels either describe the two QD states or one

QD state and the k = 0 WL state. Considering only the QD states whose level spacing is about 0.1 ωLO we find the spectral function shown in Fig. 7.14a, showing a series of satellites spaced≈0.5ωLOapart. In contrast, for the QD-WL coupling the level splitting is 0.56 ωLO for the s-shell and 0.44ωLO for the p-shell, we find the spectral function shown in Fig. 7.14b and Fig. 7.14c, showing dominant satellites spaced≈ 1 ωLO apart.

In addition to the level splitting, important differences are in the ratio of diagonal and off-diagonal coupling. From the coupling matrix elements we infer that the QD-WL coupling is approximately a factor of ten smaller, compared to the off-diagonal QD-QD coupling. In this sense the QD-WL coupling resembles the independent-boson model, cf. Chap. 5.1.3.

However, for the total spectral function the weight of the QD-WL spectral function dom-inates as an integration over the nearly dispersionless VB1 continuum is involved. An approximate superposition of both contributions to the spectral functions of the QD states is presented in Fig. 7.14d. We find that this model reproduces the main features of the RPA calculation shown in Fig. 7.13b. For the VB1 spectral function the contributions of QD-WL coupling are negligible. Because of the q12 dependency of the WL-WL interaction matrix elements the diagonal couplings dominate and hence also the main features of the independent-boson model are found.

Consequences for carrier scattering

From the structure of the spectral function, several consequences for carrier scattering can be estimated by the scattering kernel in Markov approximation,

Λαβ

dω G!α(ω)G!β(ω±ωLO). (7.15)

For details see Chap. 5.2.1. As for the CB QD states multiple phonon satellites contribute, from the overlap between the two QD spectral functions fast scattering processes are ex-pected. In contrast, capture processes are expected to be much slower, because of the larger energetic separation of the WL band edge from the QD states. For the VB states both, cap-ture and relaxation processes are expected to be fast because of the small level spacing.

However, the picture is more complicated if considering inter-subband scattering between WL states of VB1 and VB2. As can be inferred from Fig. 7.11, the assumption of band-diagonal interaction vertices only allows for polarization scattering between the sub-bands.

Particle scattering of the formfv1 (1−fv2)is not possible within our approach and would require the inclusion of off-diagonal vertices. However, scattering between both WL VBs is still possible via capture from one WL VB into the QD and subsequent scattering into the other WL VB.