Numerical modeling of nonlinear optical properties in semiconductor diamond-shaped quantum ring with

elliptical hole

Santiago Ríosa 1, Juan Pablo Naranjoa 2 and Jairo H. Marína 3

a Escuela de Física, Grupo de Materiales Cerámicos y Vítreos, Universidad Nacional de Colombia, A.A 3840, Medellín Colombia

sariosal@unal.edu.co 1, jpnaranjor@unal.edu.co 2, jhmarin@unal.edu.co 1

MotivationThe improvement in materials growth techniques have made possible practical realization of semiconductor quantum rings having different shapes. In this regard,

quantum rings with diamond-shaped an elliptical hole were recently reported [1]. Much of the present tremendous interest on these nanostructures have been

spurred by the possibility to obtain technological potential applications including opto-electronic devices and quantum architecture elements.

AFM Images of In(Ga)As/GaAs

QD’s [1]Real nano-holed system [1] Our QR model

Theoretical model- Electron’s Schrödinger Equation with effective-mass approximation for a GaAs QR, solved with finite element method:

−ℏ2

2𝑚∗∇2 + 𝑉𝑐𝑜𝑛𝑓 −

𝑖𝑒ℏ𝐵

2𝑚∗𝑥𝜕

𝜕𝑦− 𝑦

𝜕

𝜕𝑥+𝑒2𝐵2

8𝑚∗(𝑥2 + 𝑦2) + 𝑒𝐹𝑥 𝜓 𝑥, 𝑦 = 𝐸𝜓 𝑥, 𝑦 , 𝑉𝐶𝑜𝑛𝑓 = ቊ

0 , 𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑄𝑅∞ ,𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑄𝑅

- Calculation of the optical properties with the dipole moment density matrix formalism:

𝜀0𝜒1 𝜔 =

𝑁 𝑀212

𝐸21 − ℏ𝜔 − 𝑖ℏΓ12𝜀0𝜒

3 𝜔 = −𝑁 𝑀21

2𝐼𝜇𝑐

2𝑛𝑟 𝐸21 − ℏ𝜔 − 𝑖ℏΓ12∗

4 𝑀212

𝐸21 − ℏ𝜔 2 − 𝑖ℏΓ12 2−

𝑀22 −𝑀112

𝐸21 − 𝑖ℏΓ12 𝐸21 − ℏ𝜔 − 𝑖ℏΓ12

Δ𝑛 𝑖

𝑛𝑟𝜔 = 𝑅𝑒

𝜒 𝑖 𝜔

2𝑛𝑟2 , 𝑖 = 1,3 𝛼(𝑖) 𝜔 = 𝜔

𝜇

𝜀𝑟𝐼𝑚 𝜀0𝜒

𝑖 𝜔 , 𝑖 = 1,3 𝑀𝑖𝑗 = ⟨𝑖 −𝑒 Ƹ𝑟 𝑗⟩

1 2 3 4

Calculation of the relative refractive index changes under different electromagnetic fields’ and topological conditions

Magnetic field effect, varying the ratio b/a Magnetic field effect, varying the electric field

X-axis

linear

polarized

incident

photon

b/a = 2

b/a = 4,2

F = 10000 V/m

F = 20000 V/m

Circular

polarized

incident

photon

b/a = 2

b/a = 4,2

F = 10000 V/m

F = 20000 V/m

5

Calculation of the intraband optical absorption coefficient under different electromagnetic fields’ and topological conditions

Magnetic field effect, varying the ratio b/a Magnetic field effect, varying the electric field

X-axis

linear

polarized

incident

photon

b/a = 2

b/a = 4,2

F = 10000 V/m

F = 20000 V/m

Circular

polarized

incident

photon

b/a = 2

b/a = 4,2

F = 10000 V/m

F = 20000 V/m

6

Calculation of the optical properties for an eccentric QR

Magnetic field effect, varying ξx* Magnetic field effect, varying ξy

**

Relative refractive

index changes

ξx = 10 nm

ξx = 35 nm

ξy = 20 nm

ξy = 68 nm

Intraband absorption

coefficient

ξx = 10 nm

ξx = 35 nm

ξy = 20 nm

ξy = 68 nm

*Interaction with x-axis linear polarized photon

**Interaction with y-axis linear polarized photon

7

Comparisons with another alike research

Wave functions* Stark Effect Aharonov-Bohm Effect

(η = 0.75)

Aharonov-Bohm Effect

(η = 0.9)

Aharonov-Bohm Effect

(η = 1.0)

Elliptic

QR

[2]

Our

results

We show that under singular geometric conditions, our results tend to be very similar to previous results reported for concentric elliptical quantum ring [2].

*Electron’s wave functions for seven lowest energy levels, varying the electric field.

[1] Suwit Kiravittaya, Armando Rastelli and Oliver G Schmidt, Advanced quantum dot configurations, Rep. Prog. Phys. 72 (2009) 046502

[2] Doina Bejan, Cristina Stan, and Ecaterina C. Niculescu, Optical properties of an elliptic quantum ring: Eccentricity and electric field

effects, Optical Materials, 78 (2018) 207

References

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