# Spontaneous Exciton Collapse in a Strongly Flattened Ellipsoidal InSb Quantum Dot – Nanoscale Research Letters

#### ByK. G. Dvoyan, A. Karoui and B. Vlahovic

Aug 23, 2022 Let us consider an impermeable strongly flattened ellipsoidal QD (see Fig. 1.). Then, the potential energy of the CC in cylindrical coordinates can be written in the following form:

$$Uleft( {rho ,varphi ,Z} right) = left{ begin{gathered} 0,,,frac{{rho^{2} }}{{a_{1}^{2} }} + frac{{Z^{2} }}{{c_{1}^{2} }} le 1 hfill \ infty ,,,frac{{rho^{2} }}{{a_{1}^{2} }} + frac{{Z^{2} }}{{c_{1}^{2} }} > 1 hfill \ end{gathered} right.,,,,a_{1} > > c_{1}$$

(1)

where (a_{1}) and (c_{1}) are minor and major semiaxes of the ellipsoid, respectively. One needs to compare the geometric sizes of the QD with effective exciton radius of the CC in order to determine the quantum confinement regimes.

### Strong Quantum Confinement Regime

First, we solve the problem in the strong quantum confinement regime, when the condition (a_{ex} > > a_{1} > > c_{1}) takes place (see Fig. 1a), and (a_{ex}) is the effective Bohr radius of an exciton. In this approximation, the Coulomb interaction between an electron and a hole is much less than the quantum confinement energy, therefore the former can be neglected. Then, the problem reduces to the determination of CCs’ energy states independently. As noted above, the dispersion law for narrow-gap semiconductors is non-parabolic and is given in the following form [11, 24]:

$$E^{2} = P^{2} S^{2} + m_{e(h)}^{*,2} S^{4}$$

(2)

where (S sim 10^{8} sm/sec) is the parameter related with the semiconductor band gap (E_{g} = 2m_{e}^{*} S^{2}). Let us write the Klein–Gordon equation  for an ellipsoidal QD consisting of ({text{InSb}}), with an electron and hole when their Coulomb interaction is neglected:

$$sqrt {left( {P_{e}^{2} + P_{h}^{2} } right)S^{2} + left( {m_{e}^{*,2} + m_{h}^{*,2} } right)S^{4} } ,Psi left( {vec{r}_{e} ,vec{r}_{h} } right) = E,Psi left( {vec{r}_{e} ,vec{r}_{h} } right)$$

(3)

Here (P_{e(h)}) is the momentum operator of the CC (electron, hole), (m_{e(h)}^{*}) is the effective mass of the CC, and (E) is the total energy of the system. After simple transformations, equation (3) can be written as the reduced Schrödinger equation in dimensionless units:

$$left( { – frac{1}{2}nabla_{e}^{2} – frac{1}{2}nabla_{h}^{2} } right)Psi left( {vec{r}_{e} ,vec{r}_{h} } right) = varepsilon_{0} Psi left( {vec{r}_{e} ,vec{r}_{h} } right)$$

(4)

where (varepsilon_{0} = frac{{2varepsilon^{2} – varepsilon_{g}^{2} }}{{2varepsilon_{g} }},,,varepsilon = frac{E}{{E_{ex} }},,,varepsilon_{g} = frac{{E_{g} }}{{E_{ex} }}), (E_{ex} = frac{{hbar^{2} }}{{2mu a_{ex}^{2} }} = frac{{e^{2} }}{{kappa a_{ex} }}) is the effective Rydberg energy of an exciton, (kappa) is the dielectric constant of the semiconductor, (a_{ex} = frac{{kappa hbar^{2} }}{{mu e^{2} }}) is an exciton effective Bohr radius, (mu = frac{{m_{e}^{*} m_{h}^{*} }}{{m_{e}^{*} + m_{h}^{*} }}) is the reduced mass of an exciton, and (e) is the elementary charge. The wave functions (WFs) of the problem are sought in the form (Psi left( {vec{r}_{e} ,vec{r}_{h} } right) = Psi_{e} left( {vec{r}_{e} } right)Psi_{h} left( {vec{r}_{h} } right)). After separation of variables, one can obtain the following equation for the electron:

$$left( {nabla_{e}^{2} + 2varepsilon_{e} } right)Psi_{e} left( {vec{r}_{e} } right) = 0$$

(5)

The CC motion in the radial direction is much slower than in the direction (OZ) due to the geometric shape of the QD ((a_{1} > > c_{1})). Based on this, the Hamiltonian in the dimensionless variables can be represented as the sum of the Hamiltonians of the “fast” (hat{H}_{1}) and “slow” (hat{H}_{2}) subsystems [26, 27]:

$$hat{H} = hat{H}_{1} + hat{H}_{2} + Uleft( {r,varphi ,z} right)$$

(6)

where

$$hat{H}_{1} = – frac{{partial^{2} }}{{partial z^{2} }},,,,,hat{H}_{2} = – left( {frac{{partial^{2} }}{{partial r^{2} }} + frac{1}{r}frac{partial }{partial r} + frac{1}{{r^{2} }}frac{{partial^{2} }}{{partial varphi^{2} }}} right)$$

(7)

Here, (hat{H} = frac{{hat{rm H}}}{{E_{ex} }}), (r = frac{rho }{{a_{ex} }}), (z = frac{Z}{{a_{ex} }}). The WFs are sought in the form:

$$Psi_{e} left( {r,varphi ,z} right) = Ce^{imvarphi } chi left( {z;r} right)Rleft( r right)$$

(8)

where (C) is the normalization constant. For a fixed value of the coordinate (r) of the slow subsystem, the electron motion is localized in the one-dimensional potential well with an effective variable width:

$$Lleft( r right) = 2csqrt {1 – frac{{r^{2} }}{{a^{2} }}}$$

(9)

where (a = frac{{a_{1} }}{{a_{ex} }}) and (c = frac{{c_{1} }}{{a_{ex} }}) notations are introduced. First, let us solve the Schrödinger equation for the “fast” subsystem, which can be written in the form of the harmonic equation:

$$chi^{primeprime}left( {z;r} right) + varepsilon left( r right)chi left( {z;r} right) = 0$$

(10)

where (varepsilon left( r right)) is the energy of the “fast” subsystem. The solutions of Eq. (10) are given in the form:

$$chi left( {z;r} right) = sqrt {frac{2}{Lleft( r right)}} sin left( {frac{pi n}{{Lleft( r right)}}z + frac{pi n}{2}} right)$$

(11)

where (n) is the quantum number (QN) of the “fast” subsystem. One can obtain the “fast” subsystem energy from the boundary conditions (left. {chi left( {z;r} right)} right|_{{z = pm frac{Lleft( r right)}{2}}} = 0), taking into account the expression (9):

$$varepsilon left( r right) = frac{{pi^{2} n^{2} }}{{L^{2} left( r right)}},,,,,n = 1,2,…$$

(12)

For the lower levels of the energy spectrum, the electron motion is mainly localized in the region of the geometric center-of-gravity of the QD ((r < < a)). Based on this, one can expand in series (varepsilonleft( r right)):

$$varepsilon left( r right) approx varepsilon_{n}^{0} + omega_{n}^{2} r^{2}$$

(13)

where (varepsilon_{n}^{0} = frac{{pi^{2} n^{2} }}{{4c^{2} }}) and (omega_{n} = frac{pi n}{{2ac}}) notations are introduced. Now let us consider the CC motion in the “slow” subsystem, for which the expression (13) serves as an effective potential energy. The Schrödinger equation of the “slow” subsystem takes the form:

$$left( { – left( {frac{{partial^{2} }}{{partial r^{2} }} + frac{1}{r}frac{partial }{partial r} + frac{1}{{r^{2} }}frac{{partial^{2} }}{{partial varphi^{2} }}} right) + varepsilon_{n}^{0} + omega_{n}^{2} r^{2} } right)Rleft( r right)e^{imvarphi } = 2varepsilon_{e} ,Rleft( r right)e^{imvarphi }$$

(14)

After the change of a variable (xi = omega_{n} {kern 1pt} r^{2}) and (gamma = frac{{2varepsilon_{e} – varepsilon_{n}^{0} }}{{4omega_{n} }}) notation, Eq. (14) is written as

$$xi R^{primeprime}left( xi right) + R^{prime}left( xi right) + left( { – frac{{m^{2} }}{4xi } + gamma – frac{xi }{4}} right)Rleft( xi right) = 0$$

(15)

The solution of Eq. (15) is sought in the form of (Rleft( xi right) sim e^{{ – frac{xi }{2}}} xi^{{frac{left| m right|}{2}}} Omega left( xi right)), after which the Kummer equation is obtained:

$$xi ,Omega^{primeprime}left( xi right) + left( {left| m right| + 1 – xi } right)Omega^{prime}left( xi right) + left( {gamma – frac{left| m right| + 1}{2}} right)Omega left( xi right) = 0$$

(16)

the solutions of which are given by degenerate hypergeometric functions of the first kind:

$$Omega left( xi right) = {}_{1}F_{1} left( { – left( {gamma – frac{left| m right| + 1}{2}} right),left| m right| + 1,xi } right)$$

(17)

For the total energy of an electron, from the boundary conditions, one obtains

$$varepsilon_{e} = frac{{pi^{2} n^{2} }}{{8c^{2} }} + frac{pi n}{{2ac}}left( {2n_{r} + left| m right| + 1} right) = frac{{pi^{2} n^{2} }}{{8c^{2} }} + frac{pi n}{{2ac}}left( {N + 1} right)$$

(18)

where (n_{r} ,,m) and (N = 2n_{r} + left| m right|) are the radial, magnetic, and oscillatory QNs of an electron, respectively. The electron energy (18) is a constant of separation of variables in the hole reduced Schrödinger equation:

$$left( {nabla_{h}^{2} + 2left( {varepsilon_{0} – varepsilon_{e} } right)} right)Psi_{h} left( {vec{r}_{h} } right) = 0$$

(19)

Solving equation (19) in a similar way, in the strong quantum confinement regime, one can derive the following expression for the total energy of the particles’ system:

$$varepsilon_{str}^{Kane} = sqrt {varepsilon_{g} } sqrt {frac{{pi^{2} left( {n^{2} + n^{{prime}{2}} } right)}}{{8c^{2} }} + frac{pi }{2ac}left( {nleft( {2n_{r} + left| m right| + 1} right) + n^{prime}left( {2n^{prime}_{r} + left| {m^{prime}} right| + 1} right)} right) + frac{{varepsilon_{g} }}{2}} ,$$

(20)

Here, (n,n_{r} ,,m) and (n^{prime},n^{prime}_{r} ,,m^{prime}) are the QNs of electron and hole, respectively. For comparison (see (20)), in the case of a parabolic dispersion law (e.g., for QD consisting of (GaAs)) the total energy in the strong quantum confinement regime is given as :

$$varepsilon_{str}^{par} = frac{{pi^{2} n^{2} }}{{8c^{2} }} + frac{pi n}{{2ac}}left( {N + 1} right) + frac{{pi^{2} n^{{prime}{2}} }}{{8c^{2} }} + frac{{pi n^{prime}}}{2ac}left( {N^{prime} + 1} right),,,,,N,N^{prime} = 0,1,2,…$$

(21)

Here, (n,N) and (n^{prime},N^{prime}) are the QNs of “fast” and “slow” subsystems of electron and hole, respectively. Normalized WFs are given in the form:

$$begin{gathered} ,,,Psi_{eleft( h right)} left( {r,varphi ,z} right) = frac{{e^{imvarphi } }}{{sqrt {2pi } }}sqrt {frac{2}{Lleft( r right)}} sin left( {frac{pi n}{{Lleft( r right)}}z + frac{pi n}{2}} right) times hfill \ times sqrt {frac{pi n}{{ac}}} frac{{sqrt {n_{r} !} Gamma left( {left| m right| + 1} right)}}{{Gamma^{{{3 mathord{left/ {vphantom {3 2}} right. kern-nulldelimiterspace} 2}}} left( {left| m right| + 1 + n_{r} } right)}}e^{{ – frac{pi n}{{2ac}}r^{2} }} left( {frac{pi n}{{2ac}}r^{2} } right)^{{frac{left| m right|}{2}}} {}_{1}F_{1} left{ { – n_{r} ,left| m right| + 1;frac{pi n}{{2ac}}r^{2} } right}. hfill \ end{gathered}$$

(22)

### Weak Quantum Confinement Regime

Let us discuss the weak quantum confinement regime, when the condition (a_{ex} < < c_{1}) is satisfied (see Fig. 1b). Then, the binding energy of the exciton prevails over the quantum confinement energy, and the weak influence of the QD walls appears as a small correction. In other words, the quantized motion of an exciton as a whole is considered in a strongly flattened ellipsoidal QD. In the case of the presence of Coulomb interaction between an electron and hole, the Klein–Gordon equation can be written as [25, 28,29,30]

$$sqrt {left( {left( {P_{e}^{2} + P_{h}^{2} } right)S^{2} + left( {m_{e}^{*,2} + m_{h}^{*,2} } right)S^{4} } right)} Psi left( {vec{r}_{e} ,vec{r}_{h} } right) = left( {E + frac{{e^{2} }}{{kappa left| {vec{r}_{e} – vec{r}_{h} } right|}}} right)Psi left( {vec{r}_{e} ,vec{r}_{h} } right)$$

(23)

After some transformations, as in the case of a strong quantum confinement regime, the Klein–Gordon equation reduces to the Schrödinger equation with a certain effective energy. Using the coordinates of the exciton’s center-of-gravity (vec{r} = vec{r}_{e} – vec{r}_{h}), (vec{R} = frac{{m_{e}^{ * } vec{r}_{e} + m_{h}^{ * } vec{r}_{h} }}{{m_{e}^{ * } + m_{h}^{ * } }}), where (vec{r}_{e}) and (vec{r}_{h}) are the 3D radius-vectors of an electron and a hole, respectively, (m_{h}^{ * }) is the effective mass of a hole, and considering the case of a light hole (m_{e}^{ * } = m_{h}^{ * }), one can represent the system WFs in the following form:

$$Psi left( {vec{r}_{e} ,vec{r}_{h} } right) = psi_{{n_{r} ,l,q}} left( {vec{r}} right)Phi_{{n_{Gr} ,n_{R} ,M}} left( {vec{R}} right)$$

(24)

Here, the WF (psi_{{n_{r} ,l,q}} left( {vec{r}} right)) describes the relative motion of an electron and a hole, and WF (Phi_{n,m,k} left( {vec{R}} right)) describes the motion of the exciton’s center-of-gravity, where (n_{r} ,,l,,q) are the radial, orbital, and magnetic QNs of the exciton, correspondingly. After switching to the new coordinates, the reduced Schrödinger equation takes the following form:

$$left( { – frac{{hbar^{2} }}{{2M_{0} }}nabla_{{vec{R}}}^{2} – frac{{hbar^{2} }}{2mu }nabla_{{vec{r}}}^{2} } right)psi_{{n_{r} ,l,q}} left( {vec{r}} right)Phi_{{n_{Gr} ,n_{R} ,M}} left( {vec{R}} right) = left( {frac{{left( {E + frac{{e^{2} }}{{kappa left| {vec{r}} right|}}} right)^{2} – left( {m_{e}^{*,2} + m_{h}^{*,2} } right)S^{4} }}{{2m_{e}^{ * } S^{2} }}} right)psi_{{n_{r} ,l,q}} left( {vec{r}} right)Phi_{{n_{Gr} ,n_{R} ,M}} left( {vec{R}} right)$$

(25)

where (M_{0} = m_{e}^{ * } + m_{h}^{ * }) is the mass of an exciton. In the (E_{ex}) and (a_{ex}) units, Eq. (25)  is written in the form:

$$left( { – frac{1}{4}nabla_{{vec{R}}}^{2} – nabla_{{vec{r}}}^{2} } right)psi_{{n_{r} ,l,q}} left( {vec{r}} right)Phi_{{n_{Gr} ,n_{R} ,M}} left( {vec{R}} right) = left( {varepsilon_{0} + frac{alpha }{r} + frac{beta }{{r^{2} }}} right)psi_{{n_{r} ,l,q}} left( {vec{r}} right)Phi_{{n_{Gr} ,n_{R} ,M}} left( {vec{R}} right)$$

(26)

where (varepsilon_{0} = frac{{2varepsilon^{2} – varepsilon_{g}^{2} }}{{2varepsilon_{g} }},,,alpha = frac{4varepsilon }{{varepsilon_{g} }},,,beta = frac{4}{{varepsilon_{g} }}) notations are introduced. One can derive the equation for the exciton’s center-of-gravity, after separation of variables:

$$- frac{1}{4}nabla_{{vec{R}}}^{2} Phi_{{n_{Gr} ,n_{R} ,Mleft( {N_{R} } right)}} left( {vec{R}} right) = varepsilon_{Gr} Phi_{{n_{Gr} ,n_{R} ,Mleft( {N_{R} } right)}} left( {vec{R}} right)$$

(27)

The energy (varepsilon_{Gr}) of the exciton’s center-of-gravity can be obtained by repeating the procedure of calculations of the strong quantum confinement regime for the adiabatic approximation, considering the exciton mass (M_{0}) instead of the (m_{e}^{ * }):

$$varepsilon_{Gr} = frac{{pi^{2} n_{Gr}^{2} }}{{16c^{2} }} + frac{{pi n_{Gr} }}{4ac}left( {2n_{R} + left| M right| + 1} right) = frac{{pi^{2} n_{Gr}^{2} }}{{16c^{2} }} + frac{{pi n_{Gr} }}{4ac}left( {N_{R} + 1} right)$$

(28)

where (n_{Gr}) is the QN of the “fast” subsystem of exciton’s center-of-gravity motion, (n_{R} ,,M), and (N_{R} = 2n_{R} + left| M right|) are the radial, magnetic, and oscillatory QNs of the “slow” subsystem of the same motion, respectively.

Further, let us consider the relative motion of the electron–hole pair. The WFs of the problem are sought in the form (psi_{{n_{r} ,l,q}} left( {vec{r}} right) = frac{1}{sqrt r }{rm X}_{{n_{r} ,l}} left( r right)Y_{lq} left( {theta ,varphi } right)), where (Y_{lq} left( {theta ,varphi } right)) are spherical functions, (n_{r} ,,,l,,,q) are radial, orbital, and magnetic QNs of relative motion. After simple transformations, the radial part of the reduced Schrödinger equation can be written as:

$${rm X}^{primeprime}left( r right) + frac{1}{r}{rm X}^{prime}left( r right) + left( {varepsilon_{1} – frac{{left( {l + frac{1}{2}} right)^{2} – beta }}{{r^{2} }} + frac{alpha }{r}} right){rm X}left( r right) = 0$$

(29)

where (varepsilon_{1} = varepsilon_{0} – varepsilon_{Gr}). The change of variable (eta = 2sqrt { – varepsilon_{1} } r) transforms Eq. (29) to

$${rm X}^{primeprime}left( eta right) + frac{1}{eta }{rm X}^{prime}left( eta right) + left( { – frac{1}{4} – frac{{left( {l + frac{1}{2}} right)^{2} – beta }}{{eta^{2} }} + frac{delta }{eta }} right){rm X}left( eta right) = 0$$

(30)

where the parameter (delta = frac{alpha }{{2sqrt { – varepsilon_{1} } }}) is introduced. When (eta to 0), the desired solution of (30) is sought in the form ({rm X}left( {eta to 0} right) = {rm X}_{0} sim eta^{lambda }). Substituting this in Eq. (30), one gets a quadratic equation with two solutions:

$$lambda_{1,2} = mp sqrt {left( {l + frac{1}{2}} right)^{2} – beta }$$

(31)

The solution satisfying the finiteness condition of the WF is given as ({rm X}_{0} sim eta^{{sqrt {left( {l + frac{1}{2}} right)^{2} – beta } }}). When (eta to infty), equation (30) takes the form: ({rm X}^{primeprime}left( eta right) – frac{1}{4}{rm X}left( eta right) = 0). The solution satisfying the standard conditions can be written as ({rm X}left( {eta to infty } right) = {rm X}_{infty } sim e^{{ – {raise0.7exhbox{$eta$} !mathord{left/ {vphantom {eta 2}}right.kern-nulldelimiterspace} !lower0.7exhbox{$2$}}}}) [28, 29]. Thus, the solution is sought in the form:

$${rm X}left( eta right) = eta^{lambda } e^{{ – {raise0.7exhbox{eta } !mathord{left/ {vphantom {eta 2}}right.kern-nulldelimiterspace} !lower0.7exhbox{2}}}} fleft( eta right)$$

(32)

Substituting the function (32) into Eq. (30) one gets the Kummer equation :

$$eta f^{primeprime}left( eta right) + left( {2lambda + 1 – eta } right)f^{prime}left( eta right) + left( {delta – lambda – frac{1}{2}} right)fleft( eta right) = 0$$

(33)

the solutions of which are given by the first kind degenerate hypergeometric functions:

$$fleft( eta right) =_{1} F_{1} left( { – left( {delta – lambda – frac{1}{2}} right),2lambda + 1,eta } right)$$

(34)

The expression (delta – lambda – frac{1}{2}) needs to be a nonnegative integer (n_{r}) (radial QN) providing the finiteness of the WFs:

$$n_{r} = delta – lambda – frac{1}{2},,,,,,,,n_{r} = 0,1,2,….$$

(35)

From the condition (35) for the energy, one can derive the following expression in dimensionless units:

$$varepsilon_{weak}^{Kane} = – frac{{sqrt {frac{{varepsilon_{g} }}{2} + varepsilon_{Gr} } }}{{sqrt {frac{1}{{varepsilon_{g} }} + frac{4}{{varepsilon_{g}^{2} left( {n_{r} + sqrt {left( {l + frac{1}{2}} right)^{2} – frac{4}{{varepsilon_{g} }}} + frac{1}{2}} right)^{2} }}} }}$$

(36)

or

$$varepsilon_{weak}^{Kane} = – frac{{sqrt {frac{{varepsilon_{g} }}{2} + frac{{pi^{2} n_{Gr}^{2} }}{{16c^{2} }} + frac{{pi n_{Gr} }}{4ac}left( {2n_{R} + left| M right| + 1} right)} }}{{sqrt {frac{1}{{varepsilon_{g} }} + frac{4}{{varepsilon_{g}^{2} left( {n_{r} + sqrt {left( {l + frac{1}{2}} right)^{2} – frac{4}{{varepsilon_{g} }}} + frac{1}{2}} right)^{2} }}} }}$$

(37)

For comparison, in the case of parabolic dispersion law, system energy of the weak confinement regime is given by the formula

$$varepsilon_{weak}^{par} = frac{{pi^{2} n_{Gr}^{2} }}{{16c^{2} }} + frac{{pi n_{Gr} }}{4ac}left( {2n_{R} + left| M right| + 1} right) – frac{1}{{N_{C}^{2} }} = frac{{pi^{2} n_{Gr}^{2} }}{{16c^{2} }} + frac{{pi n_{Gr} }}{4ac}left( {N_{R} + 1} right) – frac{1}{{N_{C}^{2} }}$$

(38)

where (N_{C} = 1,2,…) is the Coulomb main QN of the exciton.

It is necessary to note some important results:

1. (a)

In contrast to the case of the problem of hydrogen impurities in a semiconductor with Kane’s dispersion law, considered in works [31, 32], in the case of the exciton, the instability of the ground state energy is absent. Thus, in the case of hydrogen-like impurity, the electron energy becomes unstable when (Zalpha_{0} > frac{1}{2}) ((Z) is an ordinal number, (alpha_{0}) is the fine structure constant), and the phenomenon of the particle falling into the center takes place. However, in the case of an exciton with Kane’s dispersion law, the expression (left( {l + frac{1}{2}} right)^{2} – frac{4}{{varepsilon_{g} }}) under the square root does not become negative even for the ground state with (l = 0), hence fulfillment of the condition (alpha_{0} > frac{1}{sqrt 2 }) would be necessary to obtain instability in the ground state.

2. (b)

For narrow band gap semiconductor QDs, the quantum confined motion introduces an energy term under the square root in the energy of an exciton (expressed in the center-of-gravity referential), whereas in the case of parabolic dispersion law, this energy appears as a linear expression (a simple sum).

3. (c)

The exciton energy depends only on the main QN of the Coulomb motion in the case of realization of the parabolic dispersion, whereas in the case of Kane’s dispersion law it reveals a rather complicated dependence on the radial and orbital QNs. Thus, the non-parabolicity of the charge carrier’s dispersion law leads to the removal of “random” Coulomb degeneracy in the orbital QN.

4. (d)

In the case of the implementation of the parabolic dispersion law, a family of Coulomb levels is located under each quantum confined level; with an increase in the Coulomb QN, these levels shift closer to the quantum confined level (see 38). A similar situation is observed in the case of a diamagnetic exciton, where the magnetic quantization energy level, under which the Coulomb levels are located, takes the quantum confined level role. The picture is the opposite in the case of the implementation of the Kane’s dispersion law. In a narrow-gap semiconductor, due to interband interaction, the quantum confined and Coulomb energies do not appear as additive terms in the total energy, but as a sum under the square root (see 37). In this case, with an increase in the Coulomb QNs, the levels shift deeper into the forbidden band, moving away from the quantum confined level. This consolidation of Coulomb levels resembles the behavior of acceptor levels, while in the case of a parabolic dispersion law, exciton levels behave similarly to donor levels.

### Intermediate Quantum Confinement Regime ((a > > a_{ex} > > c))

Let us discuss the intermediate-weak quantum confinement regime, when the condition (a > > a_{ex} > > c) is satisfied (see Fig. 1c). In this regime, in the (OZ)-direction, the quantum confinement significantly exceeds the Coulomb interaction of an electron and a hole, however, in the radial direction the picture is the opposite. In this case, the system’s energy for the radial motion is caused mainly by the electron–hole Coulomb interaction, and the formation of quasi-2D exciton is possible. Note that such a quantization regime was not considered in the case of a parabolic dispersion law either. Repeating the calculation procedure as in the case of a weak quantum confinement regime, we switch to the coordinates of the center-of-gravity and relative motion. Since the motion of the center-of-gravity is the motion of an electrically neutral particle, there is no competition for this motion between quantum confinement and Coulomb quantization. Again, one solves the Klein–Gordon equation, and repeat the calculations from (23) to (28). Further, let us consider in more detail the relative motion of an electron and a hole. In the case of wide-gap semiconductors, the probability of the formation of excitons with a heavy hole more often prevails over the probability of the formation of an exciton with a light hole due to the specificity of the band structure. Therefore, in the case of an intermediate quantum confinement regime, the motion of a heavier particle, a hole, is considered in the averaged field created by a faster and lighter particle, an electron. In other words, the Born–Oppenheimer approximation is used to describe the motion of a heavy hole in a potential field created by an electron. To solve this problem, the electron potential is expanded in the Taylor series at the coordinate of the hole motion. In  this case, when a light hole is considered, such an approach would be completely unjustified. Considering the condition (a_{ex} > > c_{1}), we can assert that the motion of a quasi-2D exciton is ensured under the condition (z < < r). Therefore, instead of the averaged potential, we expand the Hamiltonian of the relative motion in a series, omitting the terms proportional to (frac{{z^{2} }}{{r^{2} }} to 0). In other words, let us consider the relative motion of a quasi-2D exciton, which is confined in the perpendicular direction by the walls of the QD. Then, in cylindrical coordinates, the reduced Schrödinger equation for relative motion will be written in the form:

$$left( {frac{{partial^{2} }}{{partial z^{2} }} + frac{{partial^{2} }}{{partial r^{2} }} + frac{1}{r}frac{partial }{partial r} + frac{1}{{r^{2} }}frac{{partial^{2} }}{{partial varphi^{2} }} + left( {varepsilon_{0} – varepsilon_{Gr} } right) + frac{alpha }{r} + frac{beta }{{r^{2} }}} right)psi_{{n_{r} ,m,n_{z} }} left( {r,varphi ,z} right) = 0$$

(39)

WFs are sought in the form of

$$psi_{{n_{r} ,m,n_{z} }} left( {r,varphi ,z} right) = Ce^{imvarphi } {rm X}_{{n_{r} ,m}} left( r right)D_{{n_{z} }} left( z right)$$

(40)

where (C) is the normalization constant. After separating the variables, for the WFs and the energy of motion in the (OZ)-direction, respectively, one gets:

$$D_{{n_{z} }} left( z right) = sqrt {frac{a}{{csqrt {a^{2} – 1} }}} sin left( {frac{{pi ,a,n_{z} }}{{2csqrt {a^{2} – 1} }}z + frac{{pi n_{z} }}{2}} right),,,,n_{z} = 1,2,…$$

(41)

$$varepsilon_{z} = frac{{pi^{2} n_{z}^{2} a^{2} }}{{4c^{2} left( {a^{2} – 1} right)}}$$

(42)

Further, for the radial part, one obtains equation

$${rm X}^{primeprime}left( r right) + frac{1}{r}{rm X}^{prime}left( r right) + left( {varepsilon_{2} – frac{{m^{2} – beta }}{{r^{2} }} + frac{alpha }{r}} right){rm X}left( r right) = 0$$

(43)

where the notation (varepsilon_{2} = varepsilon_{0} – varepsilon_{Gr} – varepsilon_{z}) is introduced. The change of variable (xi = 2sqrt { – varepsilon_{2} } r) transforms Eq. (29) to

$${rm X}^{primeprime}left( xi right) + frac{1}{xi }{rm X}^{prime}left( xi right) + left( { – frac{1}{4} – frac{{m^{2} – beta }}{{xi^{2} }} + frac{delta }{xi }} right){rm X}left( xi right) = 0$$

(44)

where the similar parameter (delta = frac{alpha }{{2sqrt { – varepsilon_{2} } }}) is introduced.

At (xi to 0) the solution of (44) is sought in the form (chi left( {xi to 0} right) = chi_{0} sim xi^{lambda }). Here, in contrast to Eq. (30) the quadratic equation is obtained with the following solutions:

$$lambda_{1,2} = mp sqrt {m^{2} – beta }$$

(45)

In 2D case, the solution satisfying the condition of finiteness of the WF is given as (chi_{0} sim xi^{{sqrt {m^{2} – beta } }}). At (xi to infty), proceeding analogously to the solution of Eq. (30), one should again arrive at the equation of Kummer (33), but with different parameter (lambda). Finally, for the energy of the 2D exciton with the Kane dispersion law one gets:

$$varepsilon_{int}^{Kane} = – frac{{sqrt {frac{{varepsilon_{g} }}{2} + frac{{pi^{2} n_{Gr}^{2} }}{{16c^{2} }} + frac{{pi n_{Gr} }}{4ac}left( {2n_{R} + left| M right| + 1} right) + frac{{pi^{2} n_{z}^{2} a^{2} }}{{4c^{2} left( {a^{2} – 1} right)}}} }}{{sqrt {frac{1}{{varepsilon_{g} }} + frac{4}{{varepsilon_{g}^{2} left( {n_{r} + sqrt {m^{2} – frac{4}{{varepsilon_{g} }}} + frac{1}{2}} right)^{2} }}} }}$$

(46)

As noted above, for an exciton with a light hole, this quantization regime was not considered even in the case of the parabolic dispersion law, hence let us consider the differences from the Kane’s dispersion law in more detail. Using the coordinates of the center-of-gravity and relative motion, the Schrödinger equation for this case is written as:

$$left( { – frac{{hbar^{2} }}{{2M_{0} }}nabla_{{vec{R}}}^{2} – frac{{hbar^{2} }}{2mu }nabla_{{vec{r}}}^{2} + frac{{e^{2} }}{{kappa left| {vec{r}} right|}}} right)psi_{{n_{r} ,m,n_{z} }} left( {vec{r}} right)Phi_{{n_{Gr} ,n_{R} ,M}} left( {vec{R}} right) = E,psi_{{n_{r} ,m,n_{z} }} left( {vec{r}} right)Phi_{{n_{Gr} ,n_{R} ,M}} left( {vec{R}} right)$$

(47)

In the (E_{ex}) and (a_{ex}) units Eq. (47) will be written in the form:

$$left( { – frac{1}{4}nabla_{{vec{R}}}^{2} – nabla_{{vec{r}}}^{2} – frac{2}{{left| {vec{r}} right|}}} right)psi_{{n_{r} ,m,n_{z} }} left( {vec{r}} right)Phi_{{n_{Gr} ,n_{R} ,M}} left( {vec{R}} right) = varepsilon ,psi_{{n_{r} ,m,n_{z} }} left( {vec{r}} right)Phi_{{n_{Gr} ,n_{R} ,M}} left( {vec{R}} right)$$

(48)

Separating the variables and solving the Schrödinger equation, as in the case of the Kane’s dispersion law, for the center-of-gravity energy one obtains the result (28). However, for radial motion, in the case of the parabolic dispersion law, one gets

$$left( {frac{{partial^{2} }}{{partial z^{2} }} + frac{{partial^{2} }}{{partial r^{2} }} + frac{1}{r}frac{partial }{partial r} + frac{1}{{r^{2} }}frac{{partial^{2} }}{{partial varphi^{2} }} + left( {varepsilon_{0} – varepsilon_{Gr} } right) + frac{2}{r}} right)psi_{{n_{r} ,m,n_{z} }} left( {r,varphi ,z} right) = 0$$

(49)

Note that Eq. (49) is also obtained considering the condition (a_{ex} > > c_{1}) and omitting the terms proportional to (frac{{z^{2} }}{{r^{2} }} to 0). Again the WFs are sought in the form (40) and for the motion in the (OZ)-direction, repeating the calculation procedure, one obtains the results (41) and (42). However, for the radial part, in this case, an equation similar to the 2D Coulomb equation is obtained:

$${rm X}^{primeprime}left( r right) + frac{1}{r}{rm X}^{prime}left( r right) + left( {varepsilon_{2} – frac{{m^{2} }}{{r^{2} }} + frac{2}{r}} right){rm X}left( r right) = 0$$

(50)

the solutions of which are given by degenerate hypergeometric functions of the first kind:

$${rm X}left( xi right) = xi^{left| m right|} e^{{ – {raise0.7exhbox{xi } !mathord{left/ {vphantom {xi 2}}right.kern-nulldelimiterspace} !lower0.7exhbox{2}}}}_{1} F_{1} left( { – left( {frac{1}{{sqrt { – varepsilon_{2} } }} – left| m right| – frac{1}{2}} right),2left| m right| + 1,xi } right)$$

(51)

A similar result for the case of the parabolic dispersion law is written as:

$$varepsilon_{int}^{Par} = frac{{pi^{2} n_{Gr}^{2} }}{{16c^{2} }} + frac{{pi n_{Gr} }}{4ac}left( {2n_{R} + left| M right| + 1} right) + frac{{pi^{2} n_{z}^{2} a^{2} }}{{4c^{2} left( {a^{2} – 1} right)}} – frac{1}{{left( {N_{C} + frac{1}{2}} right)^{2} }}$$

(52)

where (N_{C} = n_{r} + left| m right|) is Coulomb principal QN for exciton.

It is also important to make the following remarks here:

1. (a)

In contrast to the 3D exciton case, all states with (m = 0) are unstable in a semiconductor with Kane’s dispersion law. It is also important that instability is the consequence not only of the dimension reduction of the sample, but also the change in the dispersion law. “The particle falling into center” or the recombination (exciton collapse) of the pair in the states with (m = 0), is the consequence of interaction of energy bands. Thus, the dimension reduction leads to the fourfold increase in the exciton ground state energy in case of parabolic dispersion law, but in the case of Kane’s dispersion law, recombination (exciton spontaneous collapse) is also possible. Note also that the presence of quantum confinement does not affect the occurrence of instability, as it exists in both the presence and absence of quantum confinement (see formulae above).

2. (b)

Consideration of the bands’ interaction removes the degeneracy of the magnetic QN. However, the twofold degeneracy of (m) of energy remains. Thus, in the case of Kane dispersion law the exciton energy depends on (m^{2}), whereas in the parabolic case it depends on (left| m right|). Due to the circular symmetry of the problem, the twofold degeneracy of energy holds in both cases of dispersion law.

3. (c)

In this quantum confinement regime, the behavior of the Coulomb levels is similar to the case of the weak quantum confinement regime. Thus, in the case of the implementation of the Kane’s dispersion law, the Coulomb interaction appears as a mixed interference term with the quantum confinement in the total energy, while in the case of a parabolic dispersion law, the Coulomb energy appears as an additive term (compare to 46 and 52).

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