Bloch Equations: Quantum Dot Active Medium

Overview: In the following, we show how to describe carrier capture and recombination in a semiconductor nano-scale quantum dot (QD) structure. We also present equations of motions related to the material polarisation induced by electron and holes captured into QDs.

The model is based on Maxwell-Bloch equations and takes into account the following effects:

• induced recombination of electrons and holes with photon emission
• scattering of carriers between discrete QD levels (section 2.1)
• scattering of carriers between a 2-D semiconductor quantum well (QW) and the discrete QD levels (section 2.2 and 2.3)

Note: The equations describing the material polarisation can be coupled with the waveguide model introduced here to simulate opto-electronic devices like lasers, optical amplifiers, absorbers, etc.

1  Quantum Dot Bloch Equations
2  Carrier Scattering and Relaxation Processes
    2.1  Intra-dot Carrier Scattering
    2.2  LO-Phonon Induced Carrier Scattering between QDs and Wetting Layer
    2.3  Auger Scattering Processes Involving 0-D and 2-D Carriers

In this section, we introduce a semi-classical approach to model light-matter interaction in a semiconductor structure. In this approach, the light fields are described by a classical wave-equation derived from Maxwell's equations:

⎡ ⎣ ∂ ∂t ± β k0 c ϵp ∂ ∂z ⎤ ⎦ ET± = ⎡ ⎣ i 2 ω0 ⎛ ⎝ c2 ϵp ⎞ ⎠ ∂2 ∂x2 + i ω0 2 δϵa ϵp Γ− β k0 c ϵp α 2 ⎤ ⎦ ET± + i ω0 2 Γ ϵ0 ϵp PT± + F(t)ex.

(1.0)

Eq. 1.0 describes the propagation of the transverse electric field E_T^± including effects like light diffraction, wave-guiding, absorbtion, and the interaction with medium via the material polarisation P_T^±.

For detail related to the notation and the derivation of this equation please consult sections: Wave-guiding and Optical Fields and QDSOA Theoretical Model. Eq. 1.0 can be integrated numerically, given that the material polarisation P_T^± can be calculated by: P_T^± = V⁻¹∑_j,i p_j,i^± , where V is the normalizing volume, and electron and hole states are labelled using the index i and j, respectively.

The quantity p_j,i^± represents the expectation value of the quantum mechanical operator related to the microscopic polarization induced by charge carriers confined to QDs. In the following, we derive an equation of motion describing the temporal evolution of p_j,i^±. The Hermitian operator related to the microscopic polarisation in dipole approximation is defined as [9]: ∧p_j,i = b_j a_i   ∆^∗_i,j , where the following notation is used:

∆i,j     dipole matrix element, (induced by an electron in state i and a hole in state j), afi,ai     electron creation, annihilation operator, bfj,bj     hole creation, annihilation operator. (1.1)

We start with Heisenberg's equation of motion for the polarisation operator, given by [5]:

∂ ∂t ^ p   j,i  = i ħ ⎡ ⎣ ^ H   , ^ p   j,i  ⎤ ⎦

(1.2)

In free-carrier approximation and assuming solely dipole interaction with the electric field, the Hamiltonian of the physical system is given by:

^ H   = ∑ i  ϵei afi ai + ∑ j  ϵhj  bfj bj − ∑ i,j  [ afi bfj   ∆i,j + bj ai   ∆∗i,j ] ·E.

(1.3)

The first two terms represent the energy of the bound QD electron and hole states, respectively. The last term in (1.3) represents the interaction between the inter-level dipoles and the electric field. Evaluating (1.2) we get:

∂ ∂t ^ p   j,i  = − i ħ ( ϵei + ϵhj) ^ p   j,i  − i ħ {∆ij ⊗∆ij∗ [ afi ai + bfj bj − 1 ] ·E},

(1.4)

where `⊗' labels the tensor product and ∆_ij ⊗∆_ij^∗ is the dipole matrix tensor. In obtaining (1.4) we have neglected terms of the form: a^f_k a_i and b^f_l b_i, representing intra-band cross-correlations. The equations describing their dynamics contain fast oscillating contributions of the form: i(ϵ^e_k − ϵ^e_i)/ħ and i(ϵ^h_l − ϵ^h_j)/ħ, justifying the omission of these terms. Assuming that the polarisation is coherent with the optical fields introduced here and that ω₀ is the optical frequency, we make the following ansatz [5]:

~ ^ p ± j,i   = ^ p   ± j,i  e±i βz − i ω0 t     and     ~ E±   = 1 2 ( E± e±i βz − i ω0 t + E∗± e±i βz + i ω0 t ) (1.5)

Introducing ansatz (1.5) into (1.4) we obtain:

∂ ∂t ^ p   ± j,i  = ⎡ ⎣ − i ħ ( ϵei + ϵhj)+ i ω0 ⎤ ⎦ ^ p   ± j,i  − i 2 ħ {∆ij ⊗∆ij∗ [ afi ai + bfj bj − 1 ] ·E± }         − i 2 ħ {∆ij ⊗∆ij∗ [ afi ai + bfj bj − 1 ] e 2 ( ±i βz + i ω0 t ) ·E∗±} (1.6)

The last term in (1.6) can be neglected (rotating wave approximation). It contains an exponential factor that oscillates at optical frequencies and will average to zero for time-scales much larger than the period of the optical field. Equation (1.6) describes the dynamics of the Heisenberg operator related to the material polarisation induced by charge carriers confined to QDs. In a similar way, starting from Heisenberg's equation of motion for the operators a^f_i a_i and b^f_j b_j and using the ansatz (1.5) one can derive [5]:

∂ ∂t afi ai = i 2 ħ ∑ j  ^ p   f,+ j,i  ·E+ + ^ p   f,− j,i  ·E− − ^ p   + j,i  ·E∗,+ − ^ p   − j,i  ·E∗,− ∂ ∂t bfj bj = i 2 ħ ∑ i  ^ p   f,+ j,i  ·E+ + ^ p   f,− j,i  ·E− − ^ p   + j,i  ·E∗,+ − ^ p   − j,i  ·E∗,− (1.7)

Equations (1.6) and (1.7) refer to time-dependent operators in the Heisenberg picture. Using the following definitions:

pj,i± = < ^ p   ± j,i  >            microscopic polarisation, (1.8) nei = < afi ai >         occupation probability of QD electron level i, nhj = < bfj bj >        occupation probability of QD hole level j,

and taking the expectation value of (1.6) and (1.7), we obtain:

∂ ∂t pj,i± = ⎡ ⎣ − i ħ ( ϵei + ϵhj)+ i ω0 ⎤ ⎦ pj,i± − i 2 ħ {∆ij ⊗∆ij∗ [ nei + nhj − 1 ] ·E±} , (1.9) ∂ ∂t nei   = i 2 ħ ∑ j  pj,i∗,+ ·E+ + pj,i∗,− ·E− − pj,i+ ·E∗,+ − pj,i− ·E∗,−, (1.10) ∂ ∂t nhj = i 2 ħ ∑ i  p   ∗,+ j,i  ·E+ + p   ∗,− j,i  ·E− − p   + j,i  ·E∗,+ − p   − j,i  ·E∗,−. (1.11)

The coupled differential equations (1.9) - (1.11) describe the dynamics of the carrier occupation probability of bound QD states and the corresponding dipole moment. Since our starting point was the Hamiltonian (1.3) in free-carrier approximation these equations do not include terms describing relaxation and dephasing processes. Moreover, equations (1.9) - (1.11) refer to one single QD. A many-body description of the QD-WL system, including Coulomb and carrier-phonon [12] interaction, would provide a self-consistent way of describing dephasing and energy shifts due to scattering processes. Additionally, it would lead to a dependence of ω_j,i^e,h on the QD level occupation [3], [6]. While such a description is beyond the scope of this model, numerical calculations have shown that, due to the strong confinement regime, Coulomb effects on the optical properties of the QD structure presented here are relatively weak [13]. In order to include carrier capture and relaxation processes, equations (1.9) - (1.11) are extended by terms describing carrier-carrier and carrier-phonon scattering, respectively. Carrier scattering leads to a fast decay of the inter-level polarisation and will be included via an effective dephasing rate [5]. In our model, the effective polarisation dephasing rate, γ_j,i, is linked to the homogeneous broadening of the QD transitions. We also take into account the spectral broadening ∆γ_j,i and spectral shift ∆ω_j,i of the QD transition due to elastic scattering with 2-D charge carriers in the wetting layer [16]. The following definitions are introduced:

ωj,ie,h = ( ϵei + ϵhj )ħ−1 , ∆γj,i = n2De  Aj,ie + n2Dh  Aj,ih, ∆ωj,i = n2De Bj,ie + n2Dh Bj,ih, (1.12)

where ω_j,i^e,h is the angular frequency of a photon emitted by recombination of an electron in QD state i and a hole in state j, n_2D^e is the electron charge density, and n_2D^h is the hole charge density in the 2-D WL. The coefficients A_j,i and B_j,i describing spectral broadening and shift, respectively, are calculated according to [16]. Starting from the differential equations (1.9) - (1.11) and including terms related to spontaneous emission and scattering processes, we obtain:

∂ ∂t pj,i± = − (γp + ∆γj,i)pj,i± + i(ωj,ie,h − ω0 − ∆ωj,i)pj,i±             − i 2 ħ {∆ij ⊗∆ij∗ [ nei + nhj − 1 ] ·E± }, (1.13) ∂ ∂t nei = i 2 ħ ∑ j  { pj,i∗,+ ·E+ + pj,i∗,− ·E− − pj,i+ ·E∗,+ − pj,i− ·E∗,− }− ∑ j  Γspi,j  nhj  nei             + ∂ ∂t  nei ⎤ ⎦ c−ph QD  + ∂ ∂t  nei ⎢ ⎢ c−c QD ↔ WL  + ∂ ∂t  nei ⎢ ⎢ c−ph QD ↔ WL  , (1.14) ∂ ∂t nhj = i 2 ħ ∑ i  ⎧ ⎨ ⎩ p   ∗,+ j,i  ·E+ + p   ∗,− j,i  ·E− − p   + j,i  ·E∗,+ − p   − j,i  ·E∗,− ⎫ ⎬ ⎭ − ∑ i  Γspi,j  nhj  nei             + ∂ ∂t  nhj ⎢ ⎢ c−ph QD  + ∂ ∂t   nhj ⎢ ⎢ c−c QD ↔ WL  + ∂ ∂t   nhj ⎢ ⎢ c−ph QD ↔ WL  , (1.15)

where Γ^sp_i,j is the rate of spontaneous emission and the last two three in both (1.14) and (1.15) refer to carrier-phonon and carrier-carrier scattering, respectively. In our theoretical model, QDs are refilled with charge carriers via carrier capture involving emission of phonons and via carrier-carrier (Auger) capture processes. Depending on the 2-D carrier density in the wetting layer, the population of QDs with carriers, and the temperature, scattering processes may lead to caputer of carriers into the QDs or excitation of carriers out of bound QD states. At sufficiently high 2-D charge carrier densities (in the range of 10¹¹  cm⁻² − 10¹²  cm⁻²), capture processes transferring charge carriers from the sourounding semiconductor medium into discrete QD states dominate.

Fast QW↔QD carrier capture and intra-dot relaxation are essential for the ground state operation of QDLs and QDSOAs (e.g the amplification of a train of short pulses in a QDSOA [14]). In our theoretical model, we describe carrier scattering between 2-D states and discrete QDs mediated by carrier-phonon and Auger processes, respectively. Depending on the operating conditions, scattering processes can also lead to the depletion of QD states. E.g. holes in excited QD states have a high probability of being ejected into 2-D WL states due to the low confinement energy. It has been shown that Auger processes represent a major non-radiative loss channel in long wavelength devices [11]. In general, carrier-phonon and Auger scattering rates depend strongly on the transition energy. An inhomogeneous broadened QD ensemble is characterized by dot-to-dot variations of the confinement energy w.r.t. the 2-D continuum and the level structure of bound states. In our approach, we calculate scattering rates for each QD in the statistical ensemble (for details see section ). In section 1.2, we present the calculation of LO-phonon mediated intra-dot scattering rates. The description of LO-phonon and Auger mediated QW↔QD scattering rates is presented in section 1.3 and 1.4, respectively.

This type of process includes scattering of charge carriers between discrete QD states. We model intra-dot scattering of carriers coupled to LO-phonon modes. The scattering rates are calculated according to [10] under the assumption that LO-phonons decay into acoustic phonons (due to an an-harmonic coupling term). The electron-phonon coupling strength in [10] is calculated for box-like QDs, though the authors point out that the final result (relaxation rate) does not depend sensitively on the coupling constant. For simplicity we use the same scattering rates to model hole-LO-phonon scattering. As seen in Fig. 1.1 the LO-phonon intra-dot scattering rates depend on the de-phasing between the transition energy and the energy of the LO-phonon. (The frequency of LO-phonons in the InGaAs alloy is taken from [1], [2], [7].)

Figure 1.1: Rate of intra-dot electron scattering (at 300 K) with emission or absorption of one 'GaAs-like' LO-phonon as a function of the transition energy.

The term describing intra-dot carrier-phonon scattering in (1.14) and (1.15) is given by [8]:

∂ ∂t  nck ⎢ ⎢ c−ph QD  = ∑ k↑ > k  { γemk,k↑ nck↑ ( 1 − nck ) − γabsk,k↑ nck ( 1 − nck↑ ) }         + ∑ k↓ < k  { γabsk,k↓ nck↓ ( 1 − nck ) − γemk,k↓ nck ( 1 − nck↓ ) }, (1.16)

where c labels electrons or holes and k is the level index. γ_em^ph(k,k_↑) and γ_abs^ph(k,k_↑) are the scattering rates for the absorption and emission of one LO-phonon, respectively. n^c_k is the occupation probability of QD state k. The first summation on the right side of (1.16) describes the relaxation of carriers from a higher level k_↑ to a lower level k, the second summation describes the excitation of carriers from a lower level k_↓ to a higher level k.

Another scattering process involving the emission or absorption of LO-phonons is the capture of carriers from the WL into the QDs or the escape of carriers from the QDs to the 2-D wetting layer. The scattering rates for this type of process are modelled using the following equation [4]:

∂ ∂t  nck ⎢ ⎢ c−ph WL  = m∗c 4 πħ2 e2 ωLO ϵ0 ⎛ ⎝ 1 ϵ∞ − 1 ϵstat ⎞ ⎠ F(EQ) ⎡ ⎣ ⎛ ⎝ nLO + 1 ⎞ ⎠ fQ ⎛ ⎝ 1−nck ⎞ ⎠ − nLO ⎛ ⎝ 1−fQ ⎞ ⎠ nck ⎤ ⎦ (1.17)

where m^∗_c is the effective mass of the 2-D carriers in the WL and ω_LO is the LO phonon frequency. ϵ₀ is the permittivity of vacuum, ϵ_∞ and ϵ_stat are the high frequency and static dielectric constants. n_LO=[exp(ħω_LO/k_BT)−1]⁻¹ is the Bose occupation probability of LO phonons. f_Q(n^c_2D,E_Q) is the Fermi occupation probability of the 2-D carriers at the transition energy: E_Q=E_QD+ħω_LO. F(E_Q) is a form-factor that depends on the transition energy and on the wave-functions of the 0-D and 2-D states.

Figure 1.2: Rate of carrier capture (due to emission of one 'GaAs-like' LO-phonon) from a 2-D WL state to a QD state as a function of the transition energy. Temperature: 300 K.

Fig. 1.2 shows the dependence of the in-scattering rates from the 2-D wetting layer to a QD state on the transition energy. The scattering rates are calculated for cylindrical QDs [4] and depend on the confinement energies of the QD states, the geometry of QDs and WL, the temperature, the 2-D charge density, and the occupation probability of the QD states. The derivation of (1.17) is based on Fermi's golden rule using the single state energies and considering scattering events involving one LO-phonon only. This is reflected in the cut-off of the relaxation rates for transition energies larger than the LO-phonon energy. A quantum kinetic treatment of carrier-phonon interaction of the QD-WL system in the polaron picture [12] shows that fast intra-dot scattering of carriers is relatively insensitive to the detuning between transition energy and LO-phonon energy. For scattering from the 2-D wetting layer to the localized QD states a stronger dependence on the detuning is found. Direct relaxation from the WL to the QD GS can still occur even for a detuning exceeding double the LO-phonon energy, but with a reduced scattering rate. Keeping the limitations in mind, simulation of QD-WL scattering of carriers can still be described by (1.17) as long as at least one excited QD electron/hole state has a confinement energy with respect to the wetting layer states of less than ħω_LO.

Carrier-carrier scattering processes with capture of one carrier into discrete QD states have been proposed as a possible relaxation path that is especially effective at high carrier densities. QD capture rates of electrons and holes due to Auger scattering (including carrier saturation effects) are modelled by the following equations [15]:

∂ ∂t nei ⎢ ⎢ c−c WL  = Cee ne2D ne2D (1 − nei) + Ceh ne2D nh2D(1 − nei) + ∑ j  Beh nh2D nei (1 − nhj) , (1.18)

∂ ∂t nhj ⎢ ⎢ c−c WL  = Chh nh2D nh2D (1 − nhj) + Che nh2D nc2D (1 − nhj) − ∑ i  Beh nh2D nei (1 − nhj) , (1.19)

where B_he is the Auger coefficient for a scattering event in which a 2-D electron interacts with a 0-D hole and is captured into a discrete QD state, whereas the hole is scattered to the WL. C_ee is the Auger coefficient related to the interaction of two electrons from the WL and subsequent capture of one of the electrons into the QD, whereas the other is scattered into a 2-D energy state of higher energy. The Auger coefficients are labelled according to the convention that the first index denotes the captured carrier and the second index denotes the carrier scattered to a 2-D state. The Auger coefficients C_ee, C_eh, B_eh, C_he, and C_hh refer to cylindrical QDs [15] and depend on the transition energy. In our model, they are calculated for every QD in the (inhomogeneously broadened) ensemble.

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