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Power absorbed from double quantum dot-metal nanoparticle hybrid system

DQD-MNP construction

The hybrid construction studied right here consists of a DQD (the QDs are in a disk form with radii ({rho }_{1}, {rho }_{2})) and a spherical MNP of radius ((a)) at interparticle distance ((R)) (see Fig. 1) embedded in a cloth with a dielectric fixed ({varepsilon }_{B}). We additionally contemplate the radius of the DQD to be a lot smaller than that of the MNP, (left({rho }_{1}, {rho }_{2}<aright)) and in addition (left(a<Rright))27,28. This technique interacts with a linearly polarized oscillating electromagnetic area (Eleft(tright)={E}_{0}cosleft(omega tright)) with ({E}_{0}) is the electrical area amplitude, and ω is the angular frequency of the utilized area. The DQD thought-about includes two QDs; every QD was an InAs QD with a disk form and peak ({h}_{d}). The sizes of the primary QD are (({h}_{d1}) = 0.1 nm, ({rho }_{1}) = 3 nm) whereas these of the second QD are (({h}_{d2}=0.15 {textual content{nm}}), ({rho }_{2}=4 {textual content{nm}})). Every QD has one conduction and valence subband. The wetting layer (WL) within the type of a quantum effectively is an InGaAs with 10 nm thickness, and their conduction and valence subbands work as reservoir states for each QDs. The construction is grown on a GaAs barrier. The dielectric fixed of the QD is represented by ({upvarepsilon }_{mathrm{s}}) whereas the native dynamic dielectric perform of the MNP is ({upvarepsilon }_{mathrm{M}}).

Determine 1
figure 1

The hybrid DQD-MNP system. The separation between the facilities of the 2 particles is R.

The Hamiltonian of the DQD-MNP system

Contemplate a hybrid DQD-MNP construction with a pump and probe fields utilized; see Fig. 2. A probe area ({mathrm{E}}_{02}left(mathrm{t}proper)=frac{{E}_{02}^{0}}{2}{e}^{-i{omega }_{02}t}+mathrm{c}.mathrm{c}.) with a frequency ({omega }_{02}) and amplitude ({E}_{02}^{0}) is utilized between (|0rangle leftrightarrow |2rangle) DQD states. Equally, a pump laser area ({mathrm{E}}_{13}left(mathrm{t}proper)=frac{{E}_{13}^{0}}{2}{e}^{-i{omega }_{13}t}+mathrm{c}.mathrm{c}.) with a frequency ({omega }_{13}) and amplitude ({E}_{13}^{0}) is utilized between (|1rangle leftrightarrow |3rangle) DQD states. The Hamiltonian of the system might be written as,

Determine 2
figure 2

Power band diagram for the DQD-MNP system with WL.

$$H={mathrm{H}}_{0}+{mathrm{H}}_{int}+{mathrm{H}}_{calm down}$$


the place is ({mathrm{H}}_{0}) the unperturbed Hamiltonian, ({mathrm{H}}_{0}=sum_{i=0}^{5}mathrm{hslash }{upomega }_{mathrm{i}}) and the comfort Hamiltonian is ({mathrm{H}}_{calm down}). On this work, the MNP-DQD interplay Hamiltonian ({H}_{int}) is

$${H}_{int}=left[begin{array}{ccc}0& {T}_{10}& {Omega }_{20} {T}_{10}& 0& {beta }_{21} begin{array}{c}{Omega }_{20} {beta }_{30} begin{array}{c}{beta }_{40} 0end{array}end{array}& begin{array}{c}{beta }_{21} {Omega }_{31} begin{array}{c}{beta }_{41} 0end{array}end{array}& begin{array}{c}0 {T}_{23} begin{array}{c}0 {beta }_{52}end{array}end{array}end{array} begin{array}{ccc}{beta }_{30}& {beta }_{40}& 0 {Omega }_{31}& {beta }_{41}& 0 begin{array}{c}{T}_{23} 0 begin{array}{c}0 {beta }_{53}end{array}end{array}& begin{array}{c}0 0 begin{array}{c}0 {beta }_{54}end{array}end{array}& begin{array}{c}{beta }_{52} {beta }_{53} begin{array}{c}{beta }_{54} 0end{array}end{array}end{array}right]+left[begin{array}{ccc}0& 0& {mathrm{G}}_{20} 0& 0& 0 begin{array}{c}{mathrm{G}}_{20} 0 begin{array}{c}0 0end{array}end{array}& begin{array}{c}0 {mathrm{G}}_{31} begin{array}{c}0 0end{array}end{array}& begin{array}{c}0 0 begin{array}{c}0 0end{array}end{array}end{array} begin{array}{ccc}0& 0& 0 {mathrm{G}}_{31}& 0& 0 begin{array}{c}0 0 begin{array}{c}0 0end{array}end{array}& begin{array}{c}0 0 begin{array}{c}0 0end{array}end{array}& begin{array}{c}0 0 begin{array}{c}0 0end{array}end{array}end{array}right]left[{rho }_{ij}right]$$


the place ({mathrm{T}}_{01}) and ({mathrm{T}}_{32}) represents the tunneling parts, ({beta }_{ij}=frac{{mathrm{A}}_{mathrm{ij}}}{2}+frac{1}{{uptau }_{t}}), with ({mathrm{A}}_{mathrm{ij}}(=frac{{{mu }_{ij}}^{2}{omega }_{ij}^{2}}{3pi hslash {varepsilon }_{s}{c}^{3}})) is the Einstein coefficient, ({uptau }_{t}) is the dipole dephasing time, ({omega }_{ij}) is the transition frequency between QD (|mathrm{i}rangle) and (|mathrm{j}rangle) states, ({G}_{ij}) is the self-interaction of the DQD, ({mu }_{ij}) is the QD transition momentum between (|mathrm{i}rangle) and (|mathrm{j}rangle) states and ({rho }_{ij}) is the DQD density matrix operator.

The whole electrical area ({(mathrm{E}}_{DQD,ij})) felt by the DQD outcomes from the superposition of the exterior area with induced polarization of the MNP area. It’s given by,

$${E}_{DQD,ij}=frac{1}{{varepsilon }_{effs}}left({E}_{ij}+frac{1}{4pi {varepsilon }_{B}}frac{{delta }_{alpha }{P}_{MNP,ij}}{{R}^{3}}proper)$$


The induced polarization of the MNP is outlined as29,

$${P}_{MNP,ij}=left(4pi {varepsilon }_{B}proper){a}^{3}{upgamma }_{M} {E}_{MNP,ij}$$


With ({upgamma }_{M}=frac{{upvarepsilon }_{mathrm{M}}left(upomega proper)-{upvarepsilon }_{mathrm{B}}}{2{upvarepsilon }_{mathrm{B}}+{upvarepsilon }_{mathrm{M}}left(upomega proper)}). The electrical area felt by the MNP ({(E}_{MNP,ij})) is the sum of the utilized area plus the sphere because of the polarization of the DQD, i.e.,3,

$${E}_{MNP,ij}=left({E}_{ij}+frac{1}{4pi {varepsilon }_{B}}frac{{delta }_{alpha }{P}_{DQD,ij}}{{R}^{3}}proper)$$


whereas the DQD polarization is as follows

$${P}_{DQD,ij}={mu }_{mathrm{ij}}left({rho }_{ij}{e}^{-i{omega }_{ij}t}+{rho }_{ij}^{*}{e}^{i{omega }_{ij}t}proper)$$


From Eqs. (3)–(6), ({E}_{DQD,ij}) turns into,

$${E}_{DQD,ij}=frac{mathrm{hslash }}{{mu }_{ij}}left[left({Omega }_{ij}+{G}_{ij}{rho }_{ij}right){e}^{-i{omega }_{ij}t}+left({Omega }_{ij}^{*}+{G}_{ij}^{*}{rho }_{ij}^{*}right){e}^{i{omega }_{ij}t}right]$$


the place (ij) represents both the efficient Rabi frequency of the probe ({Omega }_{02}) or pump ({Omega }_{13}) area, respectively, and ({Omega }_{ij}) is taken by the relation30,

$${Omega }_{ij}={Omega }_{ij}^{0}left(1+frac{{a}^{3}{upgamma }_{M} {delta }_{alpha }}{{R}^{3}}proper)$$


The primary time period of the Rabi frequency ({Omega }_{ij}^{0}(=frac{{E}_{ij}^{0}{mu }_{ij}}{2hslash {varepsilon }_{effs}})) is said to the direct coupling of the utilized area to the DQD, whereas the second time period is the sphere produced by the MNP owing to its interplay with the utilized area. The parameter ({G}_{ij}) represents the self-interaction of the DQD and is expressed as31,

$${G}_{ij}=frac{{upgamma }_{M} {a}^{3}}{4pi {varepsilon }_{B}mathrm{hslash }{R}^{6}}{left(frac{{mu }_{ij}{delta }_{alpha }}{{upvarepsilon }_{mathrm{effs}}}proper)}^{2}$$


the place ({G}_{ij}) is produced when the utilized area polarizes the DQD, which then polarizes the MNP and creates a area that interacts with the DQD3. From Eqs. (5) and (6) we have now,

$${mathrm{E}}_{mathrm{MNP},mathrm{ij}}= left(frac{{E}_{ij}^{0}}{2}+frac{1}{4uppi {upvarepsilon }_{mathrm{B}}}frac{{updelta }_{mathrm{alpha }}{upmu }_{mathrm{ij}}}{{upvarepsilon }_{mathrm{effs}}{mathrm{R}}^{3}}{uprho }_{mathrm{ij}}proper){mathrm{e}}^{-mathrm{i}{upomega }_{mathrm{ij}}mathrm{t}}+left(frac{{E}_{ij}^{0}}{2}+frac{1}{4uppi {upvarepsilon }_{mathrm{B}}}frac{{updelta }_{mathrm{alpha }}{upmu }_{mathrm{ij}}}{{upvarepsilon }_{mathrm{effs}}{mathrm{R}}^{3}}{rho }_{ij}^{*}proper){mathrm{e}}^{mathrm{i}{upomega }_{mathrm{ij}}mathrm{t}}$$



$${E}_{MNP,mathrm{ij}}={widetilde{E}}_{MNP,mathrm{ij}}{mathrm{e}}^{-mathrm{i}{upomega }_{mathrm{ij}}mathrm{t}}+{widetilde{E}}_{MNP,mathrm{ij}}^{*}{mathrm{e}}^{mathrm{i}{upomega }_{mathrm{ij}}mathrm{t}}$$


Power absorption charge

The absorption charge from the DQD-MNP system is launched as1



Relying on the utilized fields7, the absorption charge within the DQD is supplied by,

$${Q}_{mathrm{DQD}}=hslash {omega }_{02}{rho }_{00}{gamma }_{00}+hslash {omega }_{13}{rho }_{11} {gamma }_{11}$$


To calculate the power absorbed by the MNP, take the time common of the quantity integral (int J.{E}_{MNP,tot}dV) the place (mathrm{J}) is the present density and ({E}_{MNP,tot}) is the entire electrical area contained in the MNP2,

$${E}_{MNP,tot}= sum_{ij=mathrm{02,13}}frac{{mathrm{E}}_{mathrm{MNP},mathrm{ij}}}{{upvarepsilon }_{mathrm{effM}}}=sum_{ij=mathrm{02,13}}frac{{widetilde{E}}_{MNP,mathrm{ij}}}{{upvarepsilon }_{mathrm{effM}}}{mathrm{e}}^{-mathrm{i}{upomega }_{mathrm{ij}}mathrm{t}}+frac{{widetilde{E}}_{MNP,mathrm{ij}}^{*}}{{upvarepsilon }_{mathrm{effM}}} {mathrm{e}}^{mathrm{i}{upomega }_{mathrm{ij}}mathrm{t}}$$


the place ({upvarepsilon }_{mathrm{effM}}=frac{2{varepsilon }_{B}+{varepsilon }_{M}}{3{varepsilon }_{B}}). Then,

$${mathrm{P}}_{MNP,tot}=4uppi {upvarepsilon }_{mathrm{B}}upgamma {mathrm{a}}^{3} left[sum_{ij=mathrm{02,13}}frac{{widetilde{E}}_{MNP,mathrm{ij}}}{{upvarepsilon }_{mathrm{effM}}}{mathrm{e}}^{-mathrm{i}{upomega }_{mathrm{ij}}mathrm{t}}+frac{{widetilde{E}}_{MNP,mathrm{ij}}^{*}}{{upvarepsilon }_{mathrm{effM}}} {mathrm{e}}^{mathrm{i}{upomega }_{mathrm{ij}}mathrm{t}}right]$$


The present density (J) is the same as the time spinoff of the polarization (dipole second per quantity) of the MNP3,

$$mathrm{J}=frac{partial }{partial t}left(frac{{P}_{MNP,tot}}{V}proper)$$


$$mathrm{J}=frac{-iomega left(4uppi {upvarepsilon }_{mathrm{B}}proper){upgamma }_{M} {mathrm{a}}^{3}}{V}{E}_{MNP,tot}$$


the place (V) is the quantity of the MNP. Thus, the power absorption charge by the MNP is the same as1,

$${Q}_{MNP}=int J.{E}_{MNP,tot}dV$$


This offers,

$${Q}_{mathrm{MNP}}=left(4uppi {upvarepsilon }_{mathrm{B}}proper) {mathrm{a}}^{3}upomega {upgamma }_{M} |{{widetilde{E}}_{MNP,tot}|}^{2}$$


Density matrix equations of the MNP-DQD system

The equation of movement that describes the dynamics of the DQD system is written utilizing the density matrix idea as follows32,

$${rho }_{ij}^{cdot }=frac{-i}{hslash }left[H,{rho }_{ij}right]$$


With i and j refers back to the (|mathrm{i}rangle) and (|mathrm{j}rangle) states. As within the works discussing the hybrid QD-MNP system like2,31,33, utilizing Eqs. (1) and (2), the dynamical equations of the DQD system proven in Fig. 2 are listed as,

$${rho }_{00}^{cdot }=-{gamma }_{0}{rho }_{00},+,ileft[{T}_{01}left({rho }_{10}-{rho }_{01}right)+left({Omega }_{20}+{G}_{20}{rho }_{20}right)left({rho }_{20}-{rho }_{02}right)+{beta }_{30}left({rho }_{30}-{rho }_{03}right)+{beta }_{40}left({rho }_{40}-{rho }_{04}right)right]$$

$${rho }_{11}^{cdot }=-{gamma }_{1}{rho }_{11},+, ileft[{T}_{01}left({rho }_{01}-{rho }_{10}right)+{beta }_{21}left({rho }_{21}-{rho }_{12}right)+left({Omega }_{31}+{G}_{31}{rho }_{31}right)left({rho }_{31}-{rho }_{13}right)+{beta }_{41}left({rho }_{41}-{rho }_{14}right)right]$$

$${rho }_{22}^{cdot }=-{gamma }_{2}{rho }_{22} , + , ileft[left({Omega }_{20}+{G}_{20}{rho }_{20}right)left({rho }_{02}-{rho }_{20}right)+{beta }_{21}left({rho }_{12}-{rho }_{21}right)+{T}_{32}left({rho }_{32}-{rho }_{23}right)+{beta }_{52}left({rho }_{25}-{rho }_{52}right)right]$$

$${rho }_{33}^{cdot }=-{gamma }_{3}{rho }_{33}, + ,ileft[{{beta }_{30}left({rho }_{03}-{rho }_{30}right)+left({Omega }_{31}+{G}_{31}{rho }_{31}right)left({rho }_{13}-{rho }_{31}right)+T}_{32}left({rho }_{23}-{rho }_{32}right)+{beta }_{53}left({rho }_{35}-{rho }_{53}right)right]$$

$${rho }_{44}^{cdot }=-{gamma }_{4}{rho }_{44}, + ,ileft[{beta }_{40}left({rho }_{04}-{rho }_{40}right)+{beta }_{41}left({rho }_{14}-{rho }_{41}right)right]$$

$${rho }_{55}^{cdot }=-{gamma }_{5}{rho }_{55}+ileft[{beta }_{52}left({rho }_{52}-{rho }_{25}right)+{beta }_{53}left({rho }_{53}-{rho }_{35}right)+{beta }_{54}left({rho }_{54}-{rho }_{45}right)right]$$

$${rho }_{10}^{cdot }={-left({gamma }_{0}+{gamma }_{1}proper)rho }_{10} , + , ileft[{T}_{01}left({rho }_{00}-{rho }_{11}right)+{beta }_{21}{rho }_{20}+left({Omega }_{31}+{G}_{31}{rho }_{31}right){rho }_{30}+{beta }_{41}{rho }_{40}-left({Omega }_{20}+{G}_{20}{rho }_{20}right){rho }_{12} right]$$

$${rho }_{20}^{cdot }=left[-left({gamma }_{0}+{gamma }_{2}right)-i{Delta }_{20}right]{rho }_{20}+ileft[left({Omega }_{20}+{G}_{20}{rho }_{20}right)left({rho }_{00}-{rho }_{22}right)+{beta }_{21}{rho }_{10}+{T}_{32}{rho }_{30}+{beta }_{30}{rho }_{23}+{beta }_{52}{rho }_{50}{-T}_{01}{rho }_{21}{-beta }_{40}{rho }_{24}right]$$

$${rho }_{30}^{cdot }={-left({gamma }_{0}+{gamma }_{3}proper)rho }_{30}, + , ileft[{beta }_{30}left({rho }_{00}-{rho }_{33}right)+left({Omega }_{31}+{G}_{31}{rho }_{31}right){rho }_{10}+{T}_{32}{rho }_{20}{-T}_{01}{rho }_{31}-left({Omega }_{20}+{G}_{20}{rho }_{20}right){rho }_{32}{-beta }_{40}{rho }_{34}right]$$

$${rho }_{40}^{cdot }=-left({gamma }_{0}+{gamma }_{4}proper){rho }_{40}, + , ileft[{{{beta }_{40}left({rho }_{00}-{rho }_{44}right){-beta }_{41}{rho }_{10}-left({Omega }_{20}+{G}_{20}{rho }_{20}right){rho }_{42}-beta }_{30}{rho }_{43}-T}_{01}{rho }_{41}+{beta }_{40}{rho }_{34}right]$$

$${rho }_{50}^{cdot }={-left({gamma }_{0}+{gamma }_{5}proper)rho }_{50}, + , ileft[{beta }_{52}{rho }_{20}+{beta }_{53}{rho }_{30}+{beta }_{54}{rho }_{40}+left({Omega }_{20}+{G}_{20}{rho }_{20}right){rho }_{52}{{-beta }_{30}{rho }_{53}-beta }_{40}{rho }_{54}right]$$

$${rho }_{21}^{cdot }=-left({gamma }_{1}+{gamma }_{2}proper){rho }_{21}, + , ileft[{beta }_{21}left({rho }_{11}-{rho }_{22}right)-left({Omega }_{20}+{G}_{20}{rho }_{20}right){rho }_{01}-{T}_{10}{rho }_{20}+{T}_{32}{rho }_{31}-left({Omega }_{31}+{G}_{31}{rho }_{31}right){rho }_{23}-{beta }_{41}{rho }_{24}+{beta }_{25}{rho }_{51}right]$$

$$start{aligned}{rho }_{23}^{cdot }&={-left({gamma }_{2}+{gamma }_{3}proper)rho }_{23}, + , ileft[{T}_{32}left({rho }_{33}-{rho }_{22}right)+left({Omega }_{20}+{G}_{20}{rho }_{20}right){rho }_{03}right.&quadleft.-{beta }_{03}{rho }_{20}+{beta }_{21}{rho }_{13}-left({Omega }_{31}+{G}_{31}{rho }_{31}right){rho }_{21}+{beta }_{25}{rho }_{53}-{beta }_{53}{rho }_{25}right]finish{aligned}$$

$${rho }_{24}^{cdot }=-left({gamma }_{2}+{gamma }_{4}proper){rho }_{24}, + , ileft[left({Omega }_{20}+{G}_{20}{rho }_{20}right){rho }_{04}-{beta }_{04}{rho }_{20}+{beta }_{21}{rho }_{14}{-beta }_{14}{rho }_{21}+{T}_{32}{rho }_{34}++{beta }_{25}{rho }_{54}-{beta }_{54}{rho }_{25}right]$$

$$start{aligned}{rho }_{25}^{cdot }&={-left({gamma }_{2}+{gamma }_{5}proper)rho }_{25}, + , ileft[{beta }_{25}left({rho }_{55}-{rho }_{22}right)+left({Omega }_{20}+{G}_{20}{rho }_{20}right){rho }_{05}right.&quad left.+{beta }_{21}{rho }_{15}+{T}_{32}{rho }_{35}{-beta }_{35}{rho }_{23}+{beta }_{24}{rho }_{45}-{beta }_{45}{rho }_{24}+{beta }_{25}{rho }_{55}right]finish{aligned}$$

$${rho }_{31}^{cdot }=-left({gamma }_{1}+{gamma }_{3}proper){rho }_{31}, + , ileft[left({Omega }_{31}+{G}_{31}{rho }_{31}right){left({rho }_{11}-{rho }_{33}right)+{beta }_{30}rho }_{01}{-{T}_{01}{rho }_{30}+beta }_{21}{rho }_{32}+{T}_{32}{rho }_{21}{-beta }_{35}{rho }_{51}-{beta }_{41}{rho }_{34}right]$$

$${rho }_{41}^{cdot }=-left({gamma }_{1}+{gamma }_{4}proper){rho }_{41}, + , ileft[{{beta }_{41}left({rho }_{11}-{rho }_{44}right)+{beta }_{40}rho }_{01}{-{T}_{01}{rho }_{40}-beta }_{21}{rho }_{42}+{beta }_{45}{rho }_{51}-left({Omega }_{31}+{G}_{31}{rho }_{31}right){rho }_{43}right]$$

$${rho }_{34}^{cdot }=-left({gamma }_{3}+{gamma }_{4}proper){rho }_{34}, + , ileft[{{beta }_{40}rho }_{03}+{beta }_{41}{rho }_{13}-{beta }_{03}{rho }_{40}+left({Omega }_{31}+{mathrm{G}}_{31}{uprho }_{31}right){rho }_{41}-{{T}_{32}rho }_{42}right]$$

$${rho }_{35}^{cdot }=-left({upgamma }_{3}+{upgamma }_{5}proper)uprho _{35}+mathrm{i}left[{{upbeta }_{30}uprho }_{05}+{mathrm{T}}_{32}{uprho }_{25}-{upbeta }_{25}{uprho }_{32}+{upbeta }_{35}{uprho }_{55}+left({Omega }_{31}+{mathrm{G}}_{31}{uprho }_{31}right){uprho }_{15}+{upbeta }_{35}left({uprho }_{55}-{uprho }_{33}right)right]$$

$${rho }_{51}^{cdot }={-left({gamma }_{1}+{gamma }_{5}proper)rho }_{51}, + , ileft[{{beta }_{52}rho }_{21}-{T}_{01}{rho }_{50}{-left({Omega }_{31}+{G}_{31}{rho }_{31}right){rho }_{53}+beta }_{53}{rho }_{31}-{beta }_{21}{rho }_{52}+{beta }_{54}{rho }_{41}{-beta }_{41}{rho }_{41}right]$$


With the situation,

$${rho }_{00}+{rho }_{11}+{rho }_{22}+{rho }_{33}=1$$

the place ({upgamma }_{mathrm{i}}) is the comfort charge, ({Delta }_{20}) is the detuning with ({Delta }_{20}={upomega }_{2}-{upomega }_{02}), the frequency ({upomega }_{2}) is the resonant frequency of the 2nd DQD state, and ({upomega }_{02}) is the frequency distinction between (|0rangle) and (|2rangle) DQD states.

Momentum matrix parts

Calculation of the momentum matrix component ({mu }_{ij}) (for QD states i and j) of every interdot transition, along with the calculation of every WL-QD momentum matrix component ({mu }_{iw}) of every WL-QD transition is without doubt one of the important options of this work. Momenta calculation is critical due to the essential function performed by the momenta in calculating the parameters of optical properties, particularly Rabi frequencies showing in Eqs. (7), (8), and (9), along with its implicit contribution to the calculation of ({mathrm{G}}_{ij}) and ({Omega }_{ij}) seem within the density matrix equations. Taking ({mu }_{12}) for example26,

$${mu }_{12}={C}_{mn}left{{int }_{0}^{a}{J}_{m}left({p}_{1}rho proper){J}_{m}left({p}_{2}rho proper)e{rho }^{2}drho }{int }_{0}^{{h}_{d}}[mathrm{cos}left({k}_{{z}_{1}}zright)mathrm{cos}left({k}_{{z}_{2}}zright)right]dz{int }_{0}^{2pi }frac{1}{2pi }dphi$$


the place ({mathrm{C}}_{mathrm{mn}}) is the normalization fixed, ({J}_{m}left({p}_{1}rho proper)) is the Bessel perform within the QD-disk aircraft within the (rho)-direction, (p) is decided from the boundary situations on the interface between the quantum disk and the encompassing materials, (e) is the digital cost, (rho) is disk radius,({ok}_{{z}_{i}}) is the wavenumber for the QD state (|mathrm{i}rangle) within the z-direction.

For the WL-QD transition, the momentum matrix component is outlined right here with an project for the states within the band. For instance, ({mu }_{35}) is the momentum for the WL-QD transition within the VB. It’s given by34,

$${mu }_{35}=left langle {varphi }_{QD}^{j=3}bigg|eoverrightarrow{r}bigg|{varphi }_{WLv}proper rangle$$


$${mu }_{35}=bigglangle {psi }_{QD}^{j=3}bigg|ewidehat{rho }rho bigg |{psi }_{WLv} biggrangle {A}_{{QD}_{z3}}{A}_{{w}_{z5}}int mathrm{cos}left({ok}_{{z}_{v}}zright)mathrm{cos}left({ok}_{{zw}_{v}}zright)dz$$


the place ({varphi }_{QD}^{j=3}), ({varphi }_{WLv}) are the entire wavefunctions of the QD state (|3rangle) and WL VB, respectively, whereas ({psi }_{QD}^{j=3}) and ({psi }_{WLv}) are these within the (rho)-direction, ({A}_{{QD}_{z3}}), ({A}_{{w}_{z5}}) are the normalization constants of the wavefunctions within the z-direction. Outline,

$$bigglangle {varphi }_{QD}^{j=3}bigg|ewidehat{rho }rho bigg|{psi }_{WLv}biggrangle =frac{1}{{N}_{WL}}bigg[bigglangle {varphi }_{QD}^{j=3}bigg|erho bigg|{psi }_{WLv}biggrangle -sum_{i=0}^{3}bigglangle {varphi }_{QD}^{j=3}bigg|erho bigg |{varphi }_{QD}^{mathrm{i}}biggrangle bigg langle {varphi }_{QD}^{mathrm{i}}|{varphi }_{WLv}bigg rangle$$


Notice that in Eq. (25), ({mathrm{N}}_{mathrm{WL},mathrm{j}}) is the normalization fixed within the OPW26,

$${N}_{WL}=sqrt{1-{left|sum_{i}bigglangle {varphi }_{QD}^{i}bigg|{varphi }_{WL}biggrangle proper |}^{2}}$$


The summation runs over all of the DQD subbands. For the right-hand-side of Eq. (25), one has,

$$leftlangle {varphi }_{QD}^{j=3}left|erhoright |{psi }_{WLv}rightrangle =frac{{C}_{mn}|e|}{sqrt{A}}int {J}_{m,j}left(prho proper){e}^{ikrho }{rho }^{2}drho$$


$$bigglangle varphi _{{QD}}^{{j = 3}} bigg|erho bigg |varphi _{{QD}}^{{{textual content{i}} = 2}} bigg rangle = C_{{mn,j}} C_{{mn,i}} left| e proper|smallint _{0}^{{h/2}} J_{{m,j}} rho J_{{m,i}} rho drho$$


$$bigg langle {varphi }_{QD}^{mathrm{i}} bigg |{varphi }_{WLv} bigg rangle =frac{{C}_{mn}}{sqrt{A}}int {J}_{m,j}left(prho proper){e}^{ikrho }{rho }^{2}drho$$


Then, contemplating ({mu }_{14}) for example of the WL-QD transition within the CB. It’s expressed as,

$${mu }_{14}= bigg langle {varphi }_{QD}^{j=1}bigg| er bigg |{varphi }_{WLmathrm{c}} bigg rangle$$


$${mu }_{14}= bigg langle {varphi }_{QD}^{j=1} bigg |ewidehat{rho }rho bigg |{psi }_{WLmathrm{c}} bigg rangle {A}_{{QD}_{z1}}{A}_{{w}_{z4}}int mathrm{cos}left({ok}_{{z}_{c}z}proper)mathrm{cos}left({ok}_{{zw}_{c}z}proper)dz$$


$$bigg langle {varphi }_{QD}^{j=1} bigg |ewidehat{rho }rho bigg |{psi }_{WLv} bigg rangle =frac{1}{{N}_{WL,j}} bigg [ bigg langle {varphi }_{QD}^{j=1} bigg |erho bigg |{psi }_{WLv} bigg rangle -sum_{i=0}^{1} bigg langle {varphi }_{QD}^{j=1} bigg |erho bigg |{varphi }_{QD}^{mathrm{i}=0} bigg rangle bigg langle {varphi }_{QD}^{mathrm{i}=0}|{varphi }_{WLmathrm{c}} bigg rangle$$



$$bigg langle {varphi }_{QD}^{j=1} bigg |erho bigg |{varphi }_{QD}^{mathrm{i}=0} bigg rangle ={C}_{mn,j}{C}_{mn,i}left|eright|{int }_{0}^{h/2}{J}_{m,j}rho {J}_{m,i}rho drho$$


$$bigg langle {varphi }_{QD}^{mathrm{i}=0} bigg |{varphi }_{WLmathrm{c}} bigg rangle =frac{{C}_{mn}}{sqrt{A}}int {J}_{m,j}left(prho proper){e}^{ikrho }rho drho$$


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