Microscopic Semiconductor Theory

where H_carr⁰ is the single-particle contributions for conduction and valence band carriers with the energies ε^c,v_j ,

H_carr⁰ =X

j

ε^c_jc^†_jcj+X

j

ε^v_jv_j^†vj, (5.7) and the two-particle Coulomb interaction is given by [Baer et al., 2006]

H_Coul = 1 2

X

k⁰jj⁰k

(V_k^cc0jj⁰kc^†_k0c^†_jc_j0c_k +V_k^vv0jj⁰kv_k^†₀v^†_jv_j0v_k) + X

k⁰jj⁰k

V_k^cv0jj⁰kc^†_k0v_j^†v_j0c_k. (5.8)

In the above, c_j (c^†_j) and v_j (v^†_j) are fermionic operators that annihilate (create) a conduction-band carrier in the state|ji^cand a valence-band carrier in the state|ji^v, respectively. Further, the Hamiltonian of the electromagnetic field modes inside the cavity reads

H_ph =X

ξ

~ω_ξb^†_ξb_ξ, (5.9)

where bξ (b^†_ξ) is the bosonic annihilation (creation) operator of the ξth mode of the cavity.

The energy of interaction of the QDs with the electromagnetic field inside the cavity in dipole approximation can be given by:

HD=−iX

ξ,j

(gξjc^†_jvjbξ+gξjv_j^†cjbξ) + H.c., (5.10) where the approximation of equal wave-function envelopes for conduction- and valence-band states is used. Moreover, for simplicity the coupling strength gξj is assumed to be real.

The Hamiltonian given by Eq. (5.6) together with Eqs. (5.7–5.10) determines the dynamical evolution of the carrier and field operators and, in particular, the time evolution for operator expectation values.

The EoM for quantities of interest, as for example the average photon number in the cavity modes and the average electron population in the conduction and valence bands, have source terms that contain operator expectation values of higher order. In this way, the approach bears an infinite hierarchy of equations of motion for various expectation values for photon and carrier operators. To perform a consistent truncation of the equations the CE scheme is applied (for details, see chapter 4 and references therein). Namely, starting from the expectation values of the first order of photon operators, the EoM for operator expectation values are replaced by EoM for correlation functions. For example, instead of the EoM for expectation values of amplitudes of the cavity mode operatorshb^†_ξbζi, the EoM for corresponding amplitude correlation functionsδhb^†_ξbζi=hb^†_ξbζi −hb^†_ξihbζiare used. Then, to achieve a consistent classification and inclusion of correlations up to a certain order the truncation of the equations for correlation functions rather than for expectation values is performed.

In particular, in the case of a system without coherent external excitation hb^†_ξi

= hbξi= 0 and hc^†_jvj⁰i= 0 hold. Therefore, applying the rotating-wave approxima-tion here and thereafter, the EoM for amplitude correlaapproxima-tion funcapproxima-tions of the mode operators can be given by

d

dtδhb^†_ξbζi=−(κξ+κζ)δhb^†_ξbζi+X

j,q

gξjδhc^†_jvjbξi+gξjδhv^†_jcjb^†_ζi

, (5.11) where κξ is the loss rate of the ξth cavity mode and q= 1. . . N, with N being the total number of QDs. Note, that both cavity-mode amplitude correlation functions δhb^†_ξbζiand the coupled photon-assisted polarization amplitude correlationsδhv^†_jcjb^†_ξi and δhc^†_jvjbζi are classified as doublet terms in the CE scheme, i.e., they correspond to an excitation of two electrons. In the terms of the truncation operators closing the hierarchy at doublet level would correspond to the truncation operator∆^B_δ(2)^i+F/2. The equation of motion for the photon-assisted polarization amplitude correlation read [see also Eq. (B.1) in Appendix B]:

d

dtδhv_j^†c_jb^†_ξi=−i(∆_ξj−iκ_ξ−iΓ)δhv^†_jc_jb^†_ξi+g_ξjδhc^†_jc_ji(1−δhv_j^†v_ji) +X

ξ⁰

hg_ξ0jδhb^†_ξ0b_ξi(δhc^†_jc_ji −δhv_j^†v_ji) +g_ξ0jδhc^†_jc_jb^†_ξ0b_ξi −g_ξ0jδhv_j^†v_jb^†_ξ0b_ξii

, (5.12) where ∆ξj = ε^c_j −ε^v_j −~ωξ is the detuning of the ξth cavity-mode from the QD transition and Γis a phenomenological dephasing parameter describing spectral line broadening. In the case of a bimodal cavity only the cavity modes with indices ξ = 1,2are nearly resonantly coupled to the QDs. Whereas the modes withξ6= 1,2 are not within the gain spectrum of the QD ensemble or have low Q-value. Since the population of the non-lasing modes hb^†_ξbξi and the cross-correlation functions hb^†_ξb1iand hb^†_ξb2iwith ξ6= 1,2remain negligibly small, the third terms on the right-hand side of Eq. (5.12) for ξ 6= 1,2 can be effectively set equal to zero. Thus, Eq. (5.12) for ξ 6= 1,2 can be solved in the adiabatic limit yielding a time constant τnl that describes the spontaneous emission into non-lasing modes according to the Weisskopf-Wigner theory. The spontaneous emission of QDs into non-lasing modes leading to a loss of excitation is described by a β-factor defined as the ratio of the spontaneous emission rate into the lasing modes 1/τl and the total spontaneous emission rate enhanced by the Purcell effect 1/τ_sp:

β = τ_l⁻¹

τ_sp⁻¹ = τ_l⁻¹

τ_l⁻¹+τ_nl⁻¹. (5.13)

The dynamics of the carrier population of the electrons in thes-shell is given by d

dtδhc^†_scsi=−X

ξ

Re gξqδhc^†_svsbξi

+δhc^†_pcpi(1−δhc^†_scsi)τ_c⁻¹−δhc^†_scsi(1−δhv_s^†vsi)τ_nl⁻¹. (5.14) Here, the first term on the right-hand side originates from the interaction with the cavity-modes, the second term describes the relaxation of carriers from the p- to

the s-shell with a relaxation timescale τc, and the last term represents the loss of excitation into the non-lasing modes.

We assume, according to the generic QD model used throughout this thesis, that the p-shell carriers are generated at a constant pump rate p. Then, similar to Eq. (5.14), the EoM for the carrier population of the electrons in thep-shell reads:

d

dtδhc^†_pcpi=p(δhv^†_pvpi −δhc^†_pcpi)−δhc^†_pcpi(1−δhc^†_scsi)τ_c⁻¹−δhc^†_pcpi(1−δhv^†_pvpi)τ_sp⁻¹, (5.15) where the last term on the right-hand side describes spontaneous recombination of p-shell carriers. The corresponding equations for valence band carriers are relegated into Appendix B.

The form of the expression for the intensity correlation functions suggests (see Eq. (5.5)) that to exploit the statistical properties of the light emission using inten-sity correlations, a consistent treatment within the CE up to the quadruplet order is required (correspond to the truncation operator ∆^B_δ(4)^i+F/2). As in the section before we treat the carrier correlations on Hartree-Fock level (i.e.∆^F_δ(2)). In particular, the EoM for cavity-mode intensity correlations read:

d

dtδhb^†_ξb^†_ξ0bζbζ⁰i=−(κξ+κξ⁰ +κζ+κζ⁰)δhb^†_ξb^†_ξ0bζbζ⁰i +X

j

gξjδhc^†_jvjb^†_ξ0bζbζ⁰i+gξ⁰jδhc^†_jvjb^†_ξbζbζ⁰i +gζjδhv_j^†cjb^†_ξb^†_ξ0bζ⁰i+gζ⁰jδhv^†_jcjb^†_ξb^†_ξ0bζi . (5.16) The EoM for further correlation functions of the quadruplet order, which include correlation between the photon-assisted polarization and the photon number, can be found in Appendix B [see Eqs. (B.4)–(B.7)].

Results

As described above, the quadruplet order of the CE leads to a system of coupled equations [see Eqs. (5.11)–(5.12), (5.14)–(5.15), (5.16) together with Eqs. (B.1)–

(B.7)]. The system of differential equations describes the dynamics of various cor-relations between carriers and cavity modes. In particular, the method makes it possible to obtain both amplitude and intensity correlation functions of the cavity emission modes including the effects of the carrier-photon correlations.

In the ensuing section the numerical analysis of the time evolution of the emission correlation functions is presented. To relate our theory to the experimental results we estimate the number of QDs with effective gain contribution by starting with the initial density of present QDs and excluding the ones with negligible spectral and spatial overlap. Thus, it is assumed that the cavity mode field is coupled to N identical QDs. Further, we consider continuous carrier generation in the p-shell at a constant ratep as an excitation process.

To obtain a valid comparison with the experimental results we simulate the cou-pled system using standard numerical integration routines with a realistic set of pa-rameters β = 0.2, κ1= 0.03 [1/ps], κ2= 0.0318 [1/ps], Γ = 2.06 [1/ps], τsp = 50 [ps],

τc = 1 [ps] and τv = 0.5 [ps]. The number of carriers within the frequency region of interest is estimated from the total density of QDs to be N = 40. For the assumed β = 0.2 the carrier recombination is determined by the stimulated emission into the lasing modes 1 and 2 with a characteristic time scale τl=τsp/β and into the non-lasing modes with a characteristic time scale that can be found from Eq. (5.13) for the given set of parameters. Further, we assume that the cavity mode 1 is in exact resonance with the QD transition (∆1s = 0) and the mode 2 is detuned with

∆12≡ω1−ω2= ∆2s= 0.2 [1/ps]. Figure 5.11 illustrates the model for the density of states used for the comparison with the experimental results. In Fig. 5.12 we present

D(E)

E Γ

κ₁ κ₂ QDs

M₁

M₂

E₁ E₂

Figure 5.11: Illustration of the considered model for the density of states of the QDs and the modes.

the simulation results for intensity functions for the modes nξ = hb^†_ξbξi, ξ = 1,2, autocorrelation functions and crosscorrelation as a function of the pump power. Fig-ure 5.12(a) reveals, that whereas the mode 1 shows a drastic increase of emission intensity, the intensity of the emission mode 2 reaches a maximum and then slowly decreases with increasing pump power in agreement with the experimental data de-picted in Fig. 5.10(a). The calculations further show, that, again in agreement with the experimental data in Fig. 5.10(c), the dependencies of the autocorrelation func-tions for the cavity modes1and2on the pump power exhibit dramatically different behavior. As shown in Fig. 5.12(b) for low values of pump power, the autocorre-lation function is equal to 2 characteristic for the statistics of thermal light. For higher rates of the pump power, the autocorrelation function of the mode 1 drops close to the value 1 indicating the emission of coherent laser light. In contrast, the autocorrelation function of mode 2 slightly decreases at first with increasing pump powers, but for larger values of the pump power, it increases and reaches values well above 2, which is in agreement with the behavior of the autocorrelation function detected in the experiment (see Fig. 5.10(c); recall the limited temporal resolution of the HBT configuration Fig. 5.5 and[Ulrich et al., 2007]). The gain competition behavior between the modes can be approved by plotting the crosscorrelation func-tion [see Fig. 5.12(c)], that decreases to the values smaller than unity at the power pump values for which the lasing behavior of the mode1 is observed [also, compare to Fig. 5.10(d)]. Further, numerical calculations demonstrate that the observed ef-fect is independent of the modification of the spontaneous emission rate due to the many-body interaction (not shown). Note, that the discrepancy of the experimen-tal and theoretical results for the autocorrelation function of mode 2 [Figs. 5.10(c) and 5.12(b), correspondingly] and the crosscorrelation function [Figs. 5.10(d) and 5.12(c), correspondingly] at the higher pump powers is due to the presence of both

modes in each polarization direction due to the QD induced mode coupling (see sec. 5.3). The numerical simulation of the CE truncation scheme of the quadruplet

Figure 5.12: Laser characteristics calculated with the semiconductor model. (a) Intensity for the modes 1 and 2 as a function of the pump power in a log-log plot.

(b) Autocorrelation functions of the two modes. (c) Crosscorrelation between the modes 1 and 2.

order can be approved by plotting the emission mode autocorrelation functions for higher order of truncation (not shown), which demonstrates qualitatively the same behavior of the functions independent of the order of truncation. It is important to note, that since the framework of the microscopic semiconductor theory presented in this section is based on the neglection of correlation functions, the numerical results are valid in the regime when higher order correlations remain small. As it can seen from the numerical evaluation of the truncated equations, this is not the case for pump power rates exceeding2×10⁻¹[1/ps], where the correlation functions strongly increase. In essence the CE has to fail for this system since it is a method that relies on the neglection of higher order correlation functions, applied on a system that inherently drives strong correlations in the photonic subspace. To get a deeper un-derstanding of the statistical properties of the emission in the next section we will

use a different approach to gain insight into the full photon statistics.

Im Dokument Theory of many-particle correlations and optical properties of semiconductor quantum dots - photons and quantum dots (Seite 96-102)