解析锥形量子点的电子结构与线性及非线性光学性质
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本文研究了锥形量子点（conical quantum dots）的电子结构以及线性和非线性光学性质，特别是在无限势垒的情况下。研究采用了有效质量近似法（effective mass approximation）来分析锥形量子点的电子结构，这是一种在处理低维半导体系统时常用的简化方法，它考虑了量子限制效应对电子行为的影响。
作者M. Dezhkam和A. Zakery来自伊朗设拉子大学科学学院物理系，他们对锥形量子点的线性吸收系数、非线性吸收系数以及折射率进行了精确计算。他们主要关注的是锥形量子点尺寸对这些光学特性的影响，同时也探讨了入射电磁场的强度和极化对其性能的影响。
线性光学研究了光与物质的相互作用，而第三阶非线性光学则涉及在强光照射下材料对光的非线性响应，如自聚焦、自相位调制等现象。在本研究中，通过计算得到了GaAs（镓砷）锥形量子点的线性和非线性吸收系数，这有助于理解这些材料在光电子学器件中的潜在应用，例如光探测器、激光器或者光开关等。
总吸收系数和折射率是衡量量子点光学性能的关键参数，它们不仅与量子点的几何形状（如锥形）有关，还受到尺寸参数的调控。当量子点减小，电子态密度会增加，可能导致更强的吸收和更高的折射率。同时，入射光的强度和极化状态能够改变电子的激发过程，进而影响光学响应。
此外，文章指出研究结果发表于《中国光学 letters》（Chinese Optics Letters），并在2012年12月10日发布。该研究的工作接收日期为2012年5月2日，接受日期为6月20日，最终在线发布日期为10月24日。其研究工作被归类为光学科学领域（OCIS codes: 190.4720, 160.4760），并且论文的DOI（数字对象标识符）为10.3788/COL201210.121。
这篇研究提供了深入理解锥形量子点作为光控制元件的潜力，对于设计新型高性能光电子设备具有重要意义，尤其是在量子尺寸效应显著的低维半导体材料中。
COL 10(12), 121901(2012) CHINESE OPTICS LETTERS December 10, 2012
Exact investigation of the electronic structure and the linear
and nonlinear optical properties of conical quantum dots
M. Dezhkam and A. Zakery
∗
Department of Physics, College of Sciences, Shiraz University, Shiraz 71454, Iran
∗
Corresponding author: zakeri@susc.ac.ir
Received May 2, 2012; accepted June 20, 2012; posted online October 24, 2012
Intersubband linear and third-order nonlinear optical properties of conical quantum dots with infinite
barrier potential are studied. The electronic structure of conical quantum dots through effective mass
approximation is determined analytically. Linear, nonlinear, and total absorption coefficients, as well as
the refractive indices of GaAs conical d ots, are calculated. The effects of the size of the dots and of the
incident electromagnetic field are investigated. Results show that the total absorption coefficient and t he
refractive index of the dots largely depend on the size of the dots and on the intensity and polarization of
the incident electromagnetic field.
OCIS codes: 190.4720, 160.4760.
doi: 10.3788/COL201210.121901.
Quantum dots (QDs) are quasi-zero-dimension systems,
the carriers of which are confined in all the three spa-
tial dimensions. These quantum systems, which were
first studied by Esaki in 1970
[1]
, are described as “ar-
tificial atoms” because of their δ-function-like density of
states
[2]
.
Unlike bulk crystals with band energies, QDs have dis-
crete subbands because of their three-dimensional (3D)
confinements. Intersubband transitions result in physical
and optical properties and make QDs useful for infrare d
(IR) optoelectronic devices. The abs orption coefficient
(AC) and refractive index (RI) of the host material
change because of the considerably large dipole matrix
element and the low energy of intersubband transitions.
In particular, the enhancement of nonlinear optical prop-
erties is important in QDs. Nonlinear properties depend
on incident optical intensity; thus, a t high incident inten-
sities, the nonlinear properties should be co ns idered.
Researchers have recently investigated the electronic
structure and the linear and nonlinear optical prope rties
of different shapes of QDs, such as the box-shaped
[1]
,
parabolic cy linder
[3]
, lens-shaped
[4]
, spherical
[5]
, and
disc-like
[6]
QDs with finite, infinite, or Gaussian
confining potential.
Self-assembled InAs/GaAs QDs formed through
Stranski–Krastanow growth can be pyramid-shaped
[2]
.
These systems are important for laser applications
[7]
.
Thus, researchers have approximated the pyramids by
using cones to obtain the energy levels of conical quan-
tum dots (CQDs)
[8−10]
.
In this letter, a CQD with an infinite barrier potential
is considered. The Schr¨odinger equation in the e ffective
mass approximation is solved a nalytically to determine
the electronic structure of CQDs. Subsequently, the lin-
ear, third-order nonlinear, and tota l AC and RI changes
are inves tigated. Moreover, the dependencies of these op-
tical properties on QD size and on the intensity and po-
larization of the incident electromagnetic field are stud-
ied.
We consider that an electron confined in a CQD has
an infinite barrier potential, i.e., V (r) = 0 inside and
V (r) = ∞ outside the CQD. To solve this problem
analytically, we consider several assumptions, i.e., we
deem the potential outside of the QD to be infinite, al-
though in reality, the wavefunctions penetra te into the
host material
[10]
. In the effective mass approximation,
the Schr¨odinger equation for an electron inside the cone
is
−
~
2
2m
∗
∇
2
ψ(r) = Eψ(r), (1)
where ~ is the Planck’s constant divided by 2π, and, m
∗
is the electro n effective mass, E is the energ y eigenva lue,
and ψ(r) is the eigenfunction of the system. The geome-
try of the cone is shown in Fig. 1. R
′
, h, and α are the
basis radius, height, and apex angle of the cone, respec-
tively. The azimuthal symmetry facilitates the approxi-
mation of the cone by means of spherical coor dinates. To
obtain the approximation, we consider the origin of the
frame at the cone apex. The polar axis is the co ne axis.
R = (R
′2
+ h
2
)
1/2
represents the radius of the sphere, a
part of which is the cone. The area inside the CQD is
sp e cified by r < R, 0 6 ϕ 6 2π, and θ < α.
Equation (1) is rewritten by us ing spherical coordi-
nates, and is se parable due to the infinite barr ie r po-
tential. Thus, the eigenfunction ψ(r) can be written as
ψ(r, θ, ϕ) = Nf (r)g(θ)e
imϕ
, (2)
where N is the normalization consta nt. To create a
single-valued azimuthal part, m must be an integer if the
full az imuthal rang e is allowed (|m| = 0, 1, 2, · · · ).
Substituting Eq. (2) into Eq. (1) gives radia l and θ
equations. The solutions of the θ equation are associated
Legendre functions of the first and second kinds, i.e.,
P
m
ν
(cos θ) and Q
m
ν
(cos θ). As mentioned previously, we
derive cos α 6 cos θ 6 1. Q
m
ν
(cos θ) diverges at c os θ = 1
and is thus omitted
[11]
. The associated Legendr e func-
tion P
m
ν
(cos θ) is regular at cos θ = 1 and fo r |cos θ| < 1,
but singular at cos θ = −1 unless ν is an integer. In
this instance, cos θ = −1 is excluded; thus, ν is real
but nonintegral
[12]
. The spherical Bessel functions of
the first and second kinds are solutions to the radial
equation. The spher ical B e ssel functions of the second
kind, which are called “spherical Neumann functions,”
1671-7694/2012/121901(4) 121901-1
c
2012 Chinese Optics Lett ers
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