Quantum Dot Semiconductor Amplifier: Theoretical Model

Overview: In the following, we introduce a theoretical model of an opto-electronic device with an active area containing semiconductor quantum dots (QDs). Depending on the geometry and the reflectivity of the facets, the device can work as an optical amplifier or a laser. If the bias voltage is reversed the same model can be used to describe an optical absorber.

1   Theoretical Model
      1.1  Material Polarisation and QD Level Occupation Probability
      1.2  Carrier-Carrier and Carrier-Phonon Scattering
      1.3  2-D Carrier Dynamics
      1.4  Injection Current Density
      1.5  Propagation of the Optical Fields Inside the Laser Cavity
2   Simulator Layout

In this section, we briefly introduce the equations required to simulate opto-electronic devices containing QDs as optically active material. The equations presented here are used to model:

• the material polarisation (1.1),
• the carrier occupation probability of the QD states (1.2) and (1.3),
• the rates of QD carrier scattering (1.4) - (1.7),
• the dynamics of the 2-D carrier density in the semiconductor layer adjacent to the QDs (1.8) and (1.9),
• the 2-D injection density (1.10),
• the counter-propagating optical fields in the laser cavity (1.11).

Equations (1.1), (1.2), (1.3), (1.8), (1.9), and (1.11) form a system of coupled differential equations that can be integrated using the even-odd integration scheme introduced here.

The layout of a simulator based on the theoretical model is presented in section 2.

The following three equations describe the dynamics of the microscopic polarisation (1.1) and the probability that the bound QD levels are populated by electrons (1.2) and holes (1.3), respectively. These equations are derived and discussed in more detail here.

∂ ∂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± }, (1.1) ∂ ∂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  , (1.2) ∂ ∂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  , (1.3)

The term describing intra-dot carrier-phonon scattering in (1.2) and (1.3) is given by [2]:

∂ ∂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↓ ) }, (1.4)

where c labels electrons or holes and k is the QD level index. More details about intra-dot carrier scattering can be found here.

Equations (1.2) and (1.3) also include a term related to carrier-phonon scattering with carrier exchange between the 2-D semiconductor quantum well adjacent to the QDs and bound QD states. This process is discussed here and can be modelled using: [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 ⎤ ⎦ (1.5)

The terms describing QD capture rates of electrons and holes due to Auger scattering (carrier-carrier scattering) in (1.2) and (1.3) are discussed here and modelled using the following equations:

∂ ∂t nei ⎢ ⎢ c−c WL  = Cee ne2D ne2D (1 − nei) + Ceh ne2D nh2D(1 − nei) + ∑ j  Beh nh2D nei (1 − nhj) , (1.6)

∂ ∂t nhj ⎢ ⎢ c−c WL  = Chh nh2D nh2D (1 − nhj) + Che nh2D nc2D (1 − nhj) − ∑ i  Beh nh2D nei (1 − nhj) , (1.7)

The dynamics of the 2-D carriers in the quantum well adjacent to the QDs is modelled using a diffusion equation: [2]:

∂ ∂t ne2D = J − Da ⎡ ⎣ ∂2 ∂x2 + ∂2 ∂z2 ⎤ ⎦ ne2D − Γloss − nQD   ∑ i  ⎧ ⎨ ⎩ ∂ ∂t nei ⎢ ⎢ e−ph WL  + ∂ ∂t nei ⎢ ⎢ c−c WL  ⎫ ⎬ ⎭ , (1.8) ∂ ∂t nh2D = J − Da ⎡ ⎣ ∂2 ∂x2 + ∂2 ∂z2 ⎤ ⎦ nh2D − Γloss − nQD ∑ j  ⎧ ⎨ ⎩ ∂ ∂t nhj ⎢ ⎢ h−ph WL  + ∂ ∂t nhj ⎢ ⎢ c−c WL  ⎫ ⎬ ⎭ , (1.9)

where the first term in both equations describes the carrier injection with current density J. D_a is the ambi-polar diffusion coefficient [5] and n_QD is the QD sheet density. The 2-D carrier loss rate Γ_loss includes contributions due to non-radiative, spontaneous and Auger-recombination, respectively, and is given by: Γ_loss = γ_nr  n^e_2D + γ_sp  n^e_2D n^h_2D + γ_aug  n^e_2D n^e_2D n^h_2D . The last two terms in (1.8) and (1.9) describe scattering between carriers confined to QDs and 2-D charge carriers.

We describe the injection current density J in in terms of the applied bias voltage V_bias. Assuming an Ohmic regime, we have: J = σE_⊥ = σV_bias/d_⊥, where σ is the conductivity and d_⊥ is the device dimension perpendicular to the injection stripe. The conductivity is given by: σ = e (μ^en^e_3D + μ^hn^h_3D), where e is the electron charge, μ^e is the electron mobility, μ^h is the hole mobility, n^e_3D is the 3D electron charge density, and n^h_3D is the 3D hole charge density. Additionally, we approximate the 3D charge density by: n_3D = n_2D/d_⊥.

Since we include radiative and non-radiative electron-hole recombination explicitly, we write the term describing carrier injection in (1.8) and (1.9) as:

J = e ( μ   e   ne2D + μ   h   nh2D)  Vbias − (Cepot − Chpot) d2⊥ ,

(1.10)

where C^e_pot = C^e_pot(n^e_2D) and C^h_pot = C^h_pot(n^h_2D) is the chemical potential of 2-D electrons in the conduction band and 2-D holes in the valence band, respectively. Note: The chemical potential is usually denoted by the greek letter μ. Here we chose C_pot to avoid confusion with the electron and hole mobility.

The optical fields propagating inside the laser cavity are modelled using (1.34) an equation derived in the section called 'Waveguiding and Optical Fields'.

Spontaneous emission has been included in (1.2) and (1.3) as a loss channel. However, spontaneous emission also couples to the laser radiation field, inducing amplitude and phase fluctuations (noise). In order to model the impact of spontaneous emission noise on the optical fields, we add Langevin noise sources F(t) to the 2-D wave-equation (1.34):

⎡ ⎣ ∂ ∂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.

(1.11)

We have assumed that the transverse electric field is polarised in x-direction. As a result the noise term F(t) is multiplied with the unity vector in x-direction e_x. The Langevin noise term is given by [3] [4] [5]:

F(t) = 1 2 ⎡ ⎣ 1 dt ∑ i,j  Γspj,i  nhj(t)  nei(t) ⎤ ⎦ [1/2]   ⎡ ⎣ ħ ω0 ϵ0 ϵp V ⎤ ⎦ [1/2]   exp(i2πϕ),

(1.12)

where V is the volume of the optical resonator, and ϕ is a random Gaussian variable with zero mean and unity variance. The quantity √[(ħ ω₀)/(ϵ₀ ϵ_p V)] represents the 'electric field per photon' [2]. To calculate the coefficient of spontaneous emission Γ^sp_j,i, we use the approximation [6]:

Γspj,i = ωj,i3 neff 3πϵ0ħ c3 3 ∑ k=1;l=1  ek [ ∆j,i ⊗∆j,i∗ ] el,

(1.13)

where we have multiplied the dipole matrix tensor ∆_j,i ⊗∆_j,i^∗ from the left and right with the standard unity vectors and summed over all spatial coordinates. The quantity ω_j,i in 1.13 is the optical frequency corresponding to the QD transition form level i to level j, while n_eff is the effective refractive index defined as: n_eff = √ε_p and ε_p is the effective dielectric constant .

In the following, we present the layout of a device simulator based on the theoretical model introduced in the section above.

1. Open output streams/files.
2. Initialise physical parameters:
   • length, width, height, and shape of the optical cavity,
   • reflectivity of front and rear facet, wavelength, reflective index of active area and cladding layer
   • QD energy levels, QD dipole elements, coefficients related to inhomogeneous and homogeneous broadening,
   • material and geometrical parameters describing the semiconductor heterostructure (matrix,QWell,QD),
   • temperature,
   • calculate QWell wave-functions, and sub-band energy levels,
   • calculate the effective mass of carriers in the (doped) QWell structure,
   • set up numerical grid,
   • initialise inhomogeneously broadened QD ensemble,
   • calculate QD-QWell carrier scattering rates,
   • calculate (relative) effective dielectric constant and optical confinement factor,
   • initialise differential operators.
3. Set initial boundary conditions.
4. Start numerical integration loop and repeat until final time-point is reached:
   • calculate chemical potential,
   • calculate carrier scattering rates,
   • calculate injected current density,
   • integrate 2-D charge density,
   • integrate carrier occupation probability of QD states,
   • integrate material polarisation,
   • calculate random field variable F(t),
   • integrate optical fields,
   • Output result data.
5. Close output streams/files.

[1] Y. Suematsu and A.R. Adams. Handbook of Semiconductor Lasers and Photonic Integrated Circuits. Chapman and Hall,London, 1994.

[2] E. Gehrig and O. Hess. Mesoscopic spatio-temporal theory for quantum dot lasers. Physical Review A (Atomic, Molecular, and Optical Physics),65(3):033804, 2002

[3] W. Chow, S. W. Koch, and M. Sargent. Semiconductor-Laser Physics. Springer-Verlag, 1994.

[4] H. Haug and S. W. Koch. Quantum Theory Of The Optical And Electronic Properties Of Semiconductors. Singapore, World Scientific, 1998.

[5] Ahmed, M.; Yamada, M.; Saito, M.; , Numerical modeling of intensity and phase noise in semiconductor lasers, Quantum Electronics, IEEE Journal of , vol.37, no.12, pp.1600-1610, Dec 2001

[6] J. P. Loehr. Physics of Strained Quantum Well Lasers. Kluwer Academic Publishers, 1998.