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3. КВАНТОВЫЕ НАНОТЕХНОЛОГИИ
3.1. EXCITON STATES OF THE OPTICAL ELECTRONS OF DIELECTRIC NANOPARTICLES IN DIELECTRIC MATRIX
Материалы статьи были доложены на 22 Международной конференции по лазерным технологиям (ALT'14) (6-10 октября, г. Касис, Франция)
Организаторы конференции: Институт общей физики им. А.М. Прохорова РАН, Aix-Marseille University and the National Center for Scientific Research (France), Центр лазерной технологии и ма-териаловедения (Москва), Международный лазерный центр Московского государственного университета им. М.В. Ломоносова.
Kulchin Yuri N., Director, academician of RAS, Institute of Automation and Control Processes FEB RAS, Vladivostok, Russia. E-mail: [email protected]
Dzyuba Vladimir P., Chief Researcher, professor, Institute of Automation and Control Processes FEB RAS, Vladivostok, Russia E-mail: [email protected]
Amosov Andrey V., Engineer, Institute of Automation and Control Processes FEB RAS, Vladivostok, Russia. E-mail: [email protected]
Abstract: The report presents and analyzes the experimental and theoretical results showing that the dielectric nanoparticles have the exciton states of electrons with binding energy of several electron volts, which are excited by the weak optical laser radiation. Such materials with a wide spectrum of exciton states are important for creation: exciton lasers and optical emitters; receiving and emitting optical nano-antennas; control and processing information and signals; generation low-power optical solitons; optical computers etc. Index terms: dielectric nanoparticles , exciton state, optical nonlinearity
3.1. ЭКСИТОННЫЕ СОСТОЯНИЯ ОПТИЧЕСКИХ ЭЛЕКТРОНОВ ДИЭЛЕКТРИЧЕСКИХ НАНОЧАСТИЦ В ДИЭЛЕКТРИЧЕСКОЙ МАТРИЦЕ
Кульчин Юрий Николаевич, доктор физ.-мат.наук, профессор, академик РАН, Заслуженный деятель науки Российской Федерации, Почетный работник высшего профессионального образования Российской Федерации, Почетный член международного общества SPIE, Директор. Институт Автоматики и Процессов Управления ДВО РАН, Владивосток, Россия. E-mail: [email protected]
Дзюба Владимир Пименович, г.н.с., доктор физ-мат.наук, профессор. Институт Автоматики и Процессов Управления ДВО РАН, Владивосток, Россия. E-mail: [email protected]
Амосов Андрей Владимирович, инженер, Институт Автоматики и Процессов Управления ДВО РАН, Владивосток, Россия. E-mail: [email protected]
Аннотация: В данной статье представлены и проанализированы экспериментальные и теоретические результаты, показывающие наличие в диэлектрических наночастицах экситонных состояний электронов с энергией в несколько электрон-вольт, которые могут возбуждаться слабым лазерным излучением. Такие материалы с широким спектром экситонных состояний могут найти применение в экситонных лазерах и оптических излучателях, наноантеннах, генерации оптических солитонов малой мощности, оптических компьютерах и т. д.
Ключевые слова: диэлектрические наночастицы, экситонные состояния, оптическая нелинейность.
INTRODUCTION
In recent years, experimental investigations of the nonlinear optical properties of dielectric nanocomposites containing small concentrations of dielectric nanoparticles showed that they exhibit unique optical nonlinearity in low-intensity optical radiation fields [1-11]. The anomalous nonlinear optical properties are as follows: (I) Great value of
bandgap gave reason to think that the nonlinear response of nanocomposite media occurs under ultraviolet light but it is observed for visible and infrared light [1];(II) Nonlinear response occurs at radiation intensities below 1 kW/cm2 and can be observed under pulsed and cw laser modes. It reaches a maximum and disappears with increasing intensity [2-4]; (III) It takes place if transmission spectra of nano-
particles array have the broad bands of light absorption that are absent for the bulk sample [1,5,6,10]; (IV) Nonlinear optical properties take place at frequencies lying within the absorption band of light [5]. Typical dependences the nonlinear part of the absorption coefficient and refractive index on the intensity of radiation are presented in fig. 1. The intensity threshold and nature of nonlinear response depend on characteristics of nanoparticles and matrix material as well as their size and shape. It was determined that dielectric nanoparticles have nonlinear response when the matrix has a static permittivity less than that of nanoparticles. These facts are explained by photoexcitation of exciton states of the electrons of the nanoparticle. In the report presents and analyzes the experimental and theoretical results showing that the dielectric nanoparticles have the exciton states of electrons with binding energy of several electron volts, which are excited by the weak optical laser radiation [7].
MAIN PART
The existence of nonlinear optical properties in dielectric nanocomposites points out that the electronic structure of nanoparticles dispersed in dielectric matrix differs significantly from the electronic structure of the bulk sample. Differences consist, first, of formation of the allowed energy levels for the charge carrier in the bandgap because the bandgap structure is connected with a complex form of nanoparticles and high density of surface defects in the crystal structure [8].
Fig.1 Typical dependence the nonlinear part of the absorption coefficient and refractive index on the intensity of radiation. (points and curves are experimental and theoretical results, respectively).
Moreover, the electrons of nanoparticles should have broad band of exciton states (Fig.2). A contribution of exci-ton states into optical properties is substantial if their Bohr radius is comparable to nanoparticle's size (or less than the size - the weak confinement regime). In contrast to the single-particle states, the accurate description of exciton states is impossible even for spherical nanoparticles. We can investigate an influence of nanoparticles' shape and size upon exciton energy spectrum more carefully by using an exactly solvable model of a nanoparticle represented as a system of two charge carriers - electron and electron hole, which exist inside an infinitely deep potential hole limited by paraboloid of revolution and sizes of real nanoparticles [9].
We estimate exciton energy spectrum taking into account only Coulomb interaction between electron and electron hole as well as quantum size effect in the effective mass approach. This assumption is reasonable due to comparatively small sizes of nanoparticles. Exciton's wave function ^(nXp) in a parabolic coordinate system with the centre in the centre of gravity of electron-hole pair satisfies the next equation 4
f+n
di
^ df J + ônlndn
1 ô2T J„ 2 ) . +--7+21 E +-1 = 0
fn ap2 I t+n)
(1)
Eq. (1) utilizes relative units where the Plank constant and a charge are equal to 1, masses of electron nj, electron hole
«h and exciton ^ =
are chosen in accordance
me + mh
with the effective in accordance with the effective mass approach; as a length unit Bohr radius of the exci-
ton a =-
ex
£-,
h
2. Parabolic coordinates are connected
¡ e
with Cartesian
ones as it follows
cosç
y = \j¥n sin <P' z = 2 - n) • Paraboloid of revolution
around the OZ axis in parabolic coordinates system is represented with the equation n =r (smaller values of rQ
correspond to a narrower paraboloid); a surface which is a base of nanoparticle is represented with the equation £ = 2Zq + n . Coordinates of nanoparticle's surface points satisfy the equations n q = n and £ = 2zQ + n n . The corresponding boundary condition imposed upon the wave function on the surface of the potential hole has the next
form y(nQ; 2, = 2zq + r; = Q. A solution of this equation which is finite in the centre of coordinate system and zero-equal on the infinity is expressed by using degenerated hypergeometric functions F(a,\m + 1,P) . This solution has the next
*¥{n,4,j) = C-
(a+ |m| )(ß + H )
a ! ß\
( f Í-.1H + FÍ-AH + ^-Í-TnTn + H
(2)
The n number defines exciton's binding energy levels
E. =■
je
J
2h2 sin2
2 2 H s,,n
13,6eV.
There is a ground quantum state with and P=1, a=0, a n( m +l) = n0 which satisfies the boundary condition upon
the side surface of the paraboloid of revolution. In this case using (2) we have
(i m+1)2
2
\m\ + 1 l(| m\ + 1) n0 n = —:—± J-:--+7
4
m +1
and every value of quantum number m is associated with two levels. The value of
mem
h
1
m = 0 corresponds to an exciton which has a motion plane parallel to OZ. The effective masses of electron and electron hole in, e.g. Al2O3 are equal to 0.4 m0 and 6.2 m0 . Given these masses within the nanoparticle of 40 nm height and 40 nm width of a bottom the value of 0 n is equal to 13 a.u. and the binding energies of the exciton's ground state are equal to 0.07eV and -0.04eV correspondingly. Assuming the kinetic energy of exciton's motion to be negligibly small, we come to the conclusion that energy levels of the exciton lie inside the forbidden band at a ranges of 0.07eV and 0.04eV from the bottom of conduction band. The second duplet of levels corresponding to m = 1 lies at range 0.05 eV and 0.21 eV from the conduction band to the direction of forbidden band. These values for m = 2,3,4 are equal to 0.03 eV and 0.58 eV, 0.04 eV and 1.36 eV, 0.02 and 4.06 eV correspondingly. One can observe the effect of geometrical amplification of electron-hole interaction in the spectrum of the ab-ovementioned lines. The boundary condition upon the surface £ = 2z0 + n is assosiateld with excited state with P=0, a=1 and
H+1 2 "
H +1)2 + 2z„ +n 4 HI +1
One can witness in this state a dependence of energy levels on coordinate n, which varies in the range from 0 to no. The dependence demonstrates an influence of a surface shape of a potential hole upon exciton's energy spectrum.
Fig.2. Eexciton's energy spectrum into the paraboloid of revolution
Generally, the exciton's energy spectrum consists of two areas: the first one is virtually continuous and it lies next to the bottom of the conduction band, the second one is discrete and lies deep ( several eV) of in the forbidden band. Energy spectrum of free charge carriers like energy spectrum of an exciton and is not uniform ( dependent of coordinate n ).
Thus, taking into account natural and thermal broadening of levels, we can say that energy spectrum of the exciton is formed as a virtually continuous band with a width of 0.1 eV approx., adjacent from the bottom to the conduction band, and broadened discrete levels which lie within the forbidden band at the range about (1-2) eV and more from the bottom of conduction band. Energy spectrum of charge carriers in the conduction band can have discrete levels as well. The above results may be summed up as follows: we
can build up a model of a diagram of energy levels of single-and double-particles charge carriers states in dielectric na-noparticles (Fig. 3) by using Al2O3 nanoparticles as an example. This particle is of a complex shape and its sizes are equal to 40-50 nm approx. (Fig. 3). Unlike volume samples, the nanoparticles charge carriers can have an energy spectrum with broadened quantum size levels, broad band of exciton states and a subzone of allowed energies, lying within forbidden band, caused by surface and internal defects of nanoparticle's material, impurity centers etc.. It should be noted that electronic structure of nanoparticle within a matrix depends to a great extent on relationship of dielectric permittivities of materials of matrixs and nanoparticles 82.
WhenSi/ >1 the jump permittivity on nanoparticle surface
/S2
leads to reduce the exciton binding energy and density of states of the donor levels. This reduces the width and depth of the absorption band in the transmission spectra of the nanoparticles
Fig. 3. The diagram of energy levels of single- and double-particles states of charge carriers in dielectric nanoparticles and Images of Al2O3 nanoparticles obtained by using SAFM Whens1/<j the polarization charges on the surface of
/S2
the nanoparticles increases energy. exciton binding and density of states of the donor levels nanoparticles. The absorption band of light nanoparticles becomes deeper and gets blue shift. Complex nonlinear refraction index n(ra,l) of
a nanocomposite in the field of linearly polarized light, can be expressed as n(©,I) = no +
IZ4P (® .i )
® ng )2
(ra - rang )2
(3)
Ang (Q1.Q2 )
where the sum is taken over all allowed optical transitions of charge carriers in the nanoparticle with the frequency of
n
r
ra - ra
ng
ng
+
+ 1
+ r
ng
ng
ng
®ng halfwidth r^ and dipole electric moment component of the optical transitions
P In g = < n |e | g > ' APng l®'1 ) is a radiation induced population difference between states |n) and |g) is a function of the incident radiation intensity i. Value
/ \ 2nN 1 I |2 I ,12 I ^ |2 I t|2 I 312
Ang (Ql,Q2 ) = "hn-[jlPn^l + Ql^Pngl - M ) + Q2^Png1 - M )] is a monotonic rising function of radiation intensity and Q,Q2 are the parameters of the orientational order of
nanoparticles in the external field. When the light intensity is equal to zero, due to a lack of own dipole moments of nanoparticles A n ( Q 1, Q 2 ) = 0 and the refraction
index n (®,I) = n0. The real part of the refraction index responsible for light refraction is
(I / Is
(œ-œng } + rg (1 +1 / Is )
(4)
AP }
n'(m, I) = n Q +££{ 1 -
n g
An (Qi, Q2 ){p-ang) (®-®„g )2 +r,
Electronic structure of dielectric nanoparticles is characterized by broad light absorption bands, absent within volume samples, a wide energy gap, a subzone of allowed energy (exciton, doped, etc.) of electrons within forbidden band and adjacent to the bottom of conduction band and broadened quantum-size levels (minizones) [5]. Allowance of the electronic structure of nanoparticles can be made by substituting in the Eq. (4) the integration within limits from
( mn - Am1 ) to with state densitiesg1and ^correspondingly for the summation over g states. Here m n is the frequency of interzone transition from state to a quantum-size zone, Am1 - the width of allowed energies subzone within forbidden band, A m 2 -- the width of subzone of quantum size levels which corresponds to g states. For the sake
of simplicity of the expressions to obtain, let us assume the state densities and to be independent of frequency Q1 , Q2 and Tng = rn . Going from summation to integration over
frequencies in Eq. (4) we obtain
h
n(m, I) = n0 + - £ {An (Qi, Q2 )Ap0 x
giln
(ffl-K-Ao>i))2 + f2 „(1 + -)
+ g 2ln _
+ r „(1+ 1-)
TT }
where Is is the saturation intensity. In case of nanoparticles a value of r2n can be much higher than r2ng as in
case of molecules. This is supported by the fact of considerable broadening of absorption bands in nanoparticles in comparison with that of volume samples. Follows from this expression. As the radiation intensity increases, value nonli-nearity exhibits rapid growth and reaches maximum (satu-
ration) passes through a maximum, and then decays to zero. The this behavior of the nonlinearity is observed in experiments ( fig. 1). CONCLUSION
Thus of the submissions follows that in dielectric nanopar-ticles can exist the exciton state. with the width energy spectrum of a few electron volts Their density of states depends on the shape and size of the nanoparticles and the matrix material in which they are located If the dielectric permittivity of the matrix material is less than the nanoparticles, the exciton states to actively are manifested in the nonlinear optical response. By changing the size and shape of the nanoparticles and the matrix material can generate the necessary exciton state. In this way is possible to create a new non-linear optical materials having low threshold nonlinear optical properties of the optically the active materials and working materials. Such materials with a wide spectrum of exciton states are important for creation: exciton lasers and optical emitters; receiving and emitting optical nano-antennas; control and processing information and signals; generation low-power optical solitons; optical computers etc. ACKNOWLEDGEMENTS
The research is supported by the Russian Scientific Foundation (grant № 14-12-01122), the Ministry of Education and Science of the Russian Federation (contract № 02.G25.31.0116) List of reference:
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