Quantum Well States

Overview: The first part in this section introduces a general description of 2-D WL states using a finite confinement quantum well approach and applying the effective mass approximation. An implicit equation for determining the chemical potential of 2-D carriers in quasi-equilibrium is presented in section 1.1. In section 1.2, we present equations used to model the dynamics of 2-D carrier carriers in a quantum well.

1  Description of Quantum Well States
    1.1  Chemical Potential of Doped 2-D Semiconductor Structures
    1.2  2-D Carrier Density Dynamics

In a quantum well, charge carriers are confined in one spatial direction. The confinement can be achieved by enclosing a thin sheet of semiconductor material within a different semiconductor material with a larger band-gap e.g. a strained sheet of In_0.2Ga_0.8As enclosed in a GaAs matrix.

Assumptions: Due to strain the energy degeneracy at the Γ-point of the heavy hole and light hole conduction band is lifted [4]. To model scattering involving holes, we assume that the main contribution is due to scattering of heavy holes and we neglect contributions from the light hole and split-off conduction bands. We also assume that the 2-D carriers in the WL are in quasi-equilibrium and can be described by specifying the chemical potential of the conduction and valence band, respectively.

The energy values of the quantum well sub-bands are calculated using the effective mass approximation. Hereby, we follow the approach of Loehr [4]. The total wave-function is assumed to be of the form:

Ψn,kt(r) = 1 √Ω eikt ·R uc(r)fn(y),

(1.1)

where k_t is the transverse crystal momentum vector referred to the confinement direction y, R is a lattice vector, the function u_c(r) has the periodicity of the lattice and Ω is a normalisation constant. In order to determine f_n(y) , the part of the wave-function characterising the confined 2-D states with sub-band index n, the following equation has to be solved:

⎡ ⎣ − ħ2 2 d dy 1 m∗(y) d dy +V0(y) ⎤ ⎦ fn(y) = En fn(y),

(1.2)

where V₀ , the confinement potential, is given by the band offset at the semiconductor interfaces (see Fig. ), m^∗(y) is the effective mass in the different semiconductor regions and E_n is the sub-band energy. The energy dispersion relation for electrons in the conduction band is parabolic in first order in k²_t and reads [4]:

Een(kte) = Een + ħ2 | kte |2 2 m∗,en    where     1 m∗,en = ⌠ ⌡ ∞ −∞  |fe,n(y)|2 1 m∗,e(y) dy.

(1.3)

A similar dispersion equation holds for heavy holes in the valence band:

Ehn(kth) = Ehn − ħ2 | kth |2 2 m∗,hn    where     1 m∗,hn = ⌠ ⌡ ∞ −∞  |fh,n(y)|2 1 m∗,h(y) dy.

(1.4)

Taking into consideration that the confinement potential V₀(y) in (1.2) depends on the band offsets, the applied bias voltage, and the doping concentration of the WL and the surrounding semiconductor layers, the 2-D wave-functions f_n(y) can be calculated numerically. More details on this subject are found here.

In the following, we assume that electrons and holes in the 2-D layer surrounding the QDs are in quasi-equilibrium and follow the Fermi-Dirac distributions for electrons and holes, respectively [3]:

fe(E,μe) = 1 exp( E − μe kB T ) + 1         and         fh(E,μh) = 1 exp( μh − E kB T ) + 1    ' (1.5)

where μ_e / μ_h is the chemical potential of electrons/holes, k_B is the Boltzmann constant and T is the temperature. The approximation can be used since carrier-carrier and carrier-phonon scattering will lead to the relaxation of any non-equilibrium distribution to a quasi-Fermi distribution on a femtosecond time-scale [1]. Using this assumption the charge density for electrons n^e_2D and holes n^h_2D can be expressed as [4]:

ne2D = νe ∑ n=1  kB T m∗n,e πħ2 ln ⎡ ⎣ 1 + exp ⎛ ⎝ μe − En,e kB T ⎞ ⎠ ⎤ ⎦ (1.6)

nh2D = νh ∑ n=1  kB T m∗n,h πħ2 ln ⎡ ⎣ 1 + exp ⎛ ⎝ En,h − μh kB T ⎞ ⎠ ⎤ ⎦ (1.7)

where the sum runs over the total number of 2-D electron and hole sub-bands, respectively. The effective mass for each sub-band is calculated using (1.4). In the intrinsic case, the charge density of electrons equals the charge density of holes. If the semiconductor structure is doped, the condition of charge neutrality can be formulated as: n_2D^h + N_D = n_2D^e + N_A, where N_D and N_A are the concentration of ionized donors and acceptors, respectively. Inserting (1.6) and (1.7) into the charge neutrality equation and substituting the chemical potential of the valence band μ_h using: μ_e − μ_h = eV_bias, we obtain:

νh ∑ n=1  ⎧ ⎨ ⎩ kB T m∗n,h πħ2 + En,h + eVbias − μe 2 kB T +ln ⎡ ⎣ 2 cosh ⎛ ⎝ En,h + eVbias − μe 2kB T ⎞ ⎠ ⎤ ⎦ + ND ⎫ ⎬ ⎭ = νe ∑ n = 1  ⎧ ⎨ ⎩ kB T m∗n,e πħ2 + μe − En,e 2kB T ln ⎡ ⎣ 2 cosh ⎛ ⎝ μe − En,e 2kB T ⎞ ⎠ ⎤ ⎦ ⎫ ⎬ ⎭ + NA, (1.8)

where we have used the relation: ln[ 1 + exp(x)] = x/2 + ln[ 2cosh(x/2)] in order to increase the numerical accuracy. The chemical potential of the conduction band μ_e can be determined numerically using (1.8). The numerical procedure is detailed here. Finally, we can determine the charge density of the conduction and valence band, respectively using (1.6) and (1.7).

In section 1.1, we have outlined how to determine the quasi-equilibrium charge density of electrons and holes for a given 2-D semiconductor structure subject to a bias voltage.

If we apply a forward bias voltage, carriers are injected into the device and depleted by electron-hole recombination processes or carrier scattering with the surrounding semiconductor structure (in our case the QDs and the 3D bulk semiconductor medium).

The figure on the right shows the band-structure of a QD inclosed by a QWell that in turn is surrounded by bulk semiconductor material. The applied forward voltage (indicated by a slanting of the band profile) leads to an accummulation of electrons in the conduction band (CB) and holes in the valence band (VB) of the 3D bulk medium, respectively. From here carriers cascade via a series of relaxation processes (involving scattering with the lattice and scattering with other carriers) towards lower lying energy states. This is indicated by white arrows. The yellow arrow represents spontaneous electron-hole recombination with emission of a photon (wiggly red curve).

Depending on the 2-D carrier density in the WL and the population of the QDs with carriers QD⇔WL scattering may lead to carrier capture into the QDs or ejection of carriers from the QDs. At a sufficiently high 2-D charge carrier density (of the order of 10¹¹ cm⁻²) the in-scattering of charge carriers dominates.

A quasi-equilibrium steady state is reached when carrier injection and carrier loss balance. Taking into account scattering processes between QD and WL described so far (see page: Bloch Equations sections 2.2 and 2.3), the dynamics of the WL carrier density is modelled by the following set of equations: [2]:

∂ ∂t ne2D = J − Da ⎡ ⎣ ∂2 ∂x2 + ∂2 ∂z2 ⎤ ⎦ ne2D − Γloss − nQD   ∑ i  ⎧ ⎨ ⎩ ∂ ∂t nei ⎢ ⎢ e−ph WL  + ∂ ∂t nei ⎢ ⎢ c−c WL  ⎫ ⎬ ⎭ , (1.9) ∂ ∂t nh2D = J − Da ⎡ ⎣ ∂2 ∂x2 + ∂2 ∂z2 ⎤ ⎦ nh2D − Γloss − nQD ∑ j  ⎧ ⎨ ⎩ ∂ ∂t nhj ⎢ ⎢ h−ph WL  + ∂ ∂t nhj ⎢ ⎢ c−c WL  ⎫ ⎬ ⎭ , (1.10)

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 (1.9) and (1.10) describe scattering between carriers confined to QDs and 2-D charge carriers. We describe the injection current density 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 (1.9) and (1.10) as:

J = e ( - μ   e   ne2D + - μ   h   nh2D)  Vbias − (μe − μh) d2⊥ ,

(1.11)

where μ_e = μ_e(n^e_2D) and μ_h = μ_h(n^h_2D) is the chemical potential of 2-D electrons in the conduction band and 2-D holes in the valence band, respectively. Adjusting the bias voltage (as a model parameter) allows us to set the 2-D carrier density (at steady state) and implicitly the occupation probability of the QD states with charge carriers.

[1]

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

[2]

Edeltraud Gehrig and Ortwin Hess. Mesoscopic spatiotemporal theory for quantum-dot lasers. Physical Review A (Atomic, Molecular, and Optical Physics), 65(3):033804, 2002.

[3]

Richard Liboff. Introductory Quantum Mechanics. Addison-Wesley, Upper Lake River, New Jersey, fourth edition, 2003.

[4]

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

[5]

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

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