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Linear and third order nonlinear optical properties of GaAs quantum dot in terahertz region
Journal Pre-proof Linear and third order nonlinear optical properties of GaAs quantum dot in terahertz region Sukanya Nasa, S.P. Purohit PII:
S1386-9477(19)31549-8
DOI:
https://doi.org/10.1016/j.physe.2019.113913
Reference:
PHYSE 113913
To appear in:
Physica E: Low-dimensional Systems and Nanostructures
Received Date: 11 October 2019 Accepted Date: 20 December 2019
Please cite this article as: S. Nasa, S.P. Purohit, Linear and third order nonlinear optical properties of GaAs quantum dot in terahertz region, Physica E: Low-dimensional Systems and Nanostructures (2020), doi: https://doi.org/10.1016/j.physe.2019.113913. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Linear and Third Order Nonlinear Optical Properties of GaAs Quantum Dot in Terahertz Region Sukanya Nasaa) and S. P. Purohit Department of Physics and Materials Science and Engineering Jaypee Institute of Information Technology, A-10, Sector-62, Noida, 201309, India
a)
Corresponding author: [email protected]
Abstract: We study the electronic and optical properties of GaAs quantum dot embedded in Ga1-yAlyAs matrix. Using the effective mass approximation with the finite confinement potential the intraband energy levels and the wavefunctions are obtained. The linear and third order nonlinear optical properties, photoabsorption coefficient, refractive index change and susceptibility are studied for different dot radius in the terahertz region.
Keywords: Quantum dots, Third order nonlinearity, Terahertz Radiation, Nanomaterials
INTRODUCTION The physical, optical and electronic properties of GaAs have made it an important compound for various technological applications[1],[2]. The direct band gap electronic structure and the noncentrosymmetric nature of GaAs are advantageous for its optoelectronic applications. The electronic and optical properties changes to a great extent in quantum wells, quantum wires and quantum dots (QDs)[3] due to quantum confinement effects. In QD structures, the electronic properties are very similar to that of atoms. The QDs are also called artificial atoms[4]. In QDs energy levels are configurable by changing only few physical parameters like radius of the dot, size and/or shape of the dot or the change of surrounding matrix material[3]. Different experimental techniques [5], [6], [7] are reported in the literature to fabricate and study the properties of GaAs QDs. Rastelli et al.[8]
fabricated self-assembled unstrained GaAs QDs of tunable sizes by a suitable
combination of Stranski-Krastanow (SK) growth and etching. Sanguinetti et al.[9] studied the optical properties of self-assembled GaAs/GaAlAs QDs grown by modified droplet expitaxy method. Brunner et al.[10] fabricated single QDs of different sizes (250-2000 nm) by laser induced local inter-diffusion of GaAs/GaAlAs quantum wells. In recent past, theoretical studies on the linear optical properties (LOP) and nonlinear optical properties (NLOP) of GaAs nano structures have gained a considerable interest[11], [12], [13], [14],[15], [16]. Using full numeric matrix
1
diagonalization method Sahin[11] studied the third order nonlinear photoabsorption for single and double electron QDs with and without hydrogenic impurity. The Quantum Genetic Algorithm (QGA) and Hartree–Fock–Roothaan (HFR) with parabolic confinement potential are used by Cakir et al.[12] to obtain wavefunctions and energy Eigen values of different confined states to study the refractive index changes (RIC) and absorption coefficients in spherical QDs. Using the effective mass approximation (EMA) with the parabolic confinement Purohit and Mathur[17] studied the photoabsorption and photoelectric processes in GaAs QD structures. In this study we investigate the LOP and third order NLOP in terahertz region for single electron confined in spherical QD. The wave functions and energies of confined states of the conduction band are obtained by using the EMA with finite potential. The conduction band offset at the dot-matrix interface is considered as the confinement potential. Some studies on intraband optical properties of different QDs are reported in literature. Using the EMA[18][19][20] with the finite potential Anchala et al.[21] investigated the photoabsorption and photoelectric process in one electron QD and found a good agreement with experimental data[22]. “Terahertz (THz) radiation” is a term usually given to electromagnetic radiation of frequency range from 0.1 THz to 10 THz[23][24][25]. Due to its peculiar properties such as non-ionizing and non-damaging characteristics, THz covers a wide range of applications such as biomedical imaging[26], ultrafast spectroscopy[27], security[28], material identification[29] and several others[24][30]. The devices based on THz radiations are in general fast and sensitive as compared to the devices based on other regions of electromagnetic spectrum. Despite their applications in broad areas, until recently THz radiations were mostly used for spectroscopic applications. One of the main causes of its limited applicability was due to unavailability of THz sources and detectors. Recent advances in nanotechnology and optoelectronics have resulted in possibility of nano structure based THz sources and detectors. Some studies are reported in the literature for THz generation and detection from InAs and InGaAs QD structures[31][32][33][34]. The experimental and theoretical investigations are timely in demand to explore the THz sources and detectors based on GaAs nanostructures. In this study we investigate the LOP and third order NLOP like photoabsorption, RIC and susceptibility for a range of dot size in THz region.
THEORY The nonlinear optical response in terms of polarization due to incident radiation of field strength is often
described as[35],
= [ + + + ⋯ ]
where, is the n order nonlinear susceptibility and is permittivity of the free space. For a two level system
(1)
th
and are defined as [11][15][36][37] ,
2 ℇ − ℇ − ℏ − ℏγ
= and
= −
# | | 4 ' − ( % − ) 3 ℇ − ℇ − ℏ − ℏγ ℇ − ℇ − ℏ + ℏγ 'ℇ − ℇ − ℏγ('ℇ − ℇ − ℏ − ℏγ(
2
where, is carrier density in QD, ℇ is the energy of electronic state n. refers to electronic charge and f and i stands for final and initial states respectively and Tfi is the transition matrix for a transition i→f. ℏ is the energy of
incident radiation and γ is line width.
The optical absorption coefficient and the change in refractive index are the imaginary and real part of the susceptibility.
The optical absorption coefficient + is given as[36][38][39] ,
+ = ,
/0[ ] 4 .
where, - and . are permeability and real part of permittivity of the QD respectively. The linear and third order
nonlinear absorption coefficient are given by, + =
1 5'ℇ − ℇ − ℏ( 5 23 4
and + = −
− 2 # 1 / # 5 'ℇ − ℇ − ℏ( 71 − 23 4 ℏγ 4 ×
: ℇ − ℇ − ℏ − ℏγ + 2'ℇ − ℇ ('ℇ − ℇ − ℏ(; < 6
ℇ − ℇ + ℏγ
where, 5 is the delta function, 23 is the refractive index of the dot material, c is speed of light and I is intensity of incident radiation.
The 5 function is defined as
5=
ℏγ
1: ℇ − ℇ − ℏ + ℏγ ;
7
The total photoabsorption coefficient + ? is defined as,
+ ? = + + + 8 The susceptibility is related to the RIC by
∆2 = B C D 9 23 223 The corresponding linear and third order nonlinear RIC are given by
ℇ − ℇ − ℏ ∆2 = C D 10 23 2 23
ℇ − ℇ − ℏ + ℏγ
and
' − ( ∆2 # -4/ = − G4'ℇ − ℇ − ℏ( − 23 4 23 : ℇ − ℇ − ℏ + ℏγ ;
ℇ − ℇ + ℏγ × H'ℇ − ℇ − ℏ( × :'ℇ − ℇ ( × 'ℇ − ℇ − ℏ( − ℏγ ;
− ℏγ '2'ℇ − ℇ ( − ℏ(IJ 11
3
The total RIC
∆2 ? K 23 is defined as
∆2 ? ∆2 ∆2 = + 12 23 23 23 The transition matrix element is defined as,
= ℱMΨ O. ̂Ψ R 13
where, ℱ is local field factor which relates the electric field inside and outside of the dot[40]. ̂ is unit polarization
vector of the linearly polarized light.
Using the EMA, wavefunctions of the QD states ΨST O, θ, W = BS O YST θ, φ are obtained by solving
Schrodinger equation, Z−
ℏ ∇ + ]^ O _ ΨST O, θ, W = ΨST O, θ, W 14 20∗
where, ]^ O is the finite potential at the interface of the dot and matrix. BS O and YST θ, φ are radial and
angular parts of the wave function respectively and n, l and m are quantum numbers. The radial part of the wave function is considered in terms of the spherical Bessel (`S ) and modified spherical Bessel function (aS ) as, B,S = b
`S cO deO O < B gaS hO deO O > B
(15)
where, P and Q are the normalization constant. R is the dot radius and parameters ζ and η are given by[40] c=, and
∗ 20 ℇS 16 ℏ
h = ,j k
∗ 2]^ 0 − c l 17 ℏ
∗ ∗ where, 0 is the effective mass of electron inside the dot and 0mno is the effective mass of electron in the matrix
material. ρ is the ratio
∗ 0 p0∗ . Applying the appropriate boundary conditions the following transcendental mno
equation is obtained which is solved numerically jc
aSq hB r + 1 `Sq cB
j − 1 18 = −h + `S cB aS hB B The effective finite potential ]^ is given by,
]^ = b
0 , deO O < B 19 ] + ]s , deO O ≥ B
where, V0 is the conduction band offset at the dot matrix interface. Vs is the self-energy given by[40]
1 1 1 ]s = C − D + ∆]s 20 2 u3 mno u3 41u B
where,
4
∆]s ≈ 0.466
u3 − u3 mno C D 21 41u u3 B u3 − u3 mno
We use the conduction band offset and dielectric constant for the mole fraction of aluminium y=0.3 as reported by Adachi[41][42]. The values of different parameters used in the present study are given in Table 1. The atomic units are used throughout. Table 1. Values of parameters used in calculations
Parameter u3
u3 mno =13.18-3.12y
Value 13.18 12.244
∗ 0
0.067 mx
V =0.6(1.155y+0.37y2) eV
0.277 eV
Effective Bohr radius a0*
10.4 nm
Effective Hartree energy Ht*
10.5 meV
∗ 0mno =(0.067+0.083y)me
0.0919 mx
RESULTS AND DISCUSSIONS In this study we investigate the LOP and NLOP in THz region associated with intraband transitions in singly charged spherical QD of GaAs embedded in Ga1-yAlyAs. Figure 1 shows the radial part of the wavefunctions for 1s, 1p, 2s and 2p discrete energy states of QD of radius 15.6 nm.
Fig. 1. Radial wavefunctions of some energy states
5
Figure 2(a) shows various discrete energy levels of l=0, 1, 2 and 3 in the conduction band of the dot radius of 15.6 nm. Figure 2(b) shows the variation of transition energies of (1s-1p), (1p-1d), (1d-1f), (1p-2s), (2s-2p) and (2p2d) dipole transitions for the dot radius range from 10.4 nm to 30 nm. The transition energy decreases on increasing the dot radius. The transition energy in THz range is marked for the dot radius range from 10.4 nm to 30 nm.
Fig. 2(a). Discrete energy levels in the dot of radius 15.6 nm
Fig. 2(b). Variation of transition energies with dot radius
To explore the effect of changing the molar fraction of Al in the surrounding matrix we also estimate the (1s-1p) transition energy for three values of y, 0.2, 0.3 and 0.4 at dot radius R=15.6 nm. The (1s-1p) transition energy is obtained as 18.215 meV, 19.049 and 20.303 meV for the molar fraction of 0.2, 0.3 and 0.4 respectively. The change in the transition energy is only up to about 2 meV.
Figure 3 shows the total photoabsorption spectra + ? (Eqn. 8) consisting of all dipole transitions for QDs of
radius 15.6 nm and at intensity of incident radiation I= 20 MW/m2. To predict the total photoabsorption spectra in experimental
situation
with
an
assembly
of
QDs
the
effect
of
ratio
of
probability[43]
ℇ − ℇ s K z Kz = {| }− ~ a of QDs in their initial state i at temperature T with respect to the ground state 1s s
6
is included. a is the Boltzmann constant. Factor of the ratio of probability is taken at room temperature T=300K.
The photoabsorption peaks are obtained up to the incident photon energy of 220 meV. It is found that the absorption peaks in THz region are obtained at 19 meV, 23.9 meV, 28.5 meV, 35.5 meV, 36.8 meV and 41.3 meV
corresponding to (1s-1p), (1p-1d), (1d-1f), (1p-2s), (2s-2p) and (2p-2d) transitions respectively. Transitions corresponding to all absorption peaks are marked in the figure. The overlapping of absorption peaks corresponding to (1p-3s) and (2s-2p); (2d-2f) and (1d-2p); and (1f-2d) and (2d-3p) transitions is due to their approximately equal transition energies. In absorption spectra peaks corresponding to all dipole transitions are obtained. The height of different absorption peaks is found changing maximum for the 1s-1p transition to minimum for the 1d-3p transition. This is due to the different value of the transition matrix element for different transitions and the ratio of probability of the initial state. The highest absorption peak in the THz range corresponds to the 1s-1p transition and we further investigate the LOP and third order NLOP viz; the photoabsorption, RIC and the susceptibility in THz region. We investigate the effect of change in dot size and change in intensity of incident radiation on the NLOP.
Fig. 3. Variation of total photoabsorption coefficient with incident photon energy
Figure 4 shows the change in the linear absorption, the nonlinear absorption and the total absorption coefficients with the variation of incident photon energy at three different dot radii, 15.6 nm, 20.8 nm and 26 nm respectively with three values of intensities 20 MW/m2, 30 MW/m2 and 40 MW/m2. The photoabsorption peaks are observed at
7
19 meV, 11.5 meV and 7.5 meV for the dot radius R 15.6 nm, 20.8 nm and 26 nm respectively. It is observed that on increasing the dot radius (Fig. 3 a, b and c) photoabsorption peak shifts towards lower energy values. This can be understood by considering the dependence of photoabsorption coefficient in inversely proportional manner to the volume of dot. At a fixed dot radius and increasing the intensity of the incident radiation (Fig. 3 a, d and g), the peak height decreases. This is due to the negative nonlinear contribution, which increases with increasing intensity.
Fig. 4. The variation of photoabsorption coefficient with incident photon energy
Figure 5 shows the RIC with the variation of incident photon energy. Results are shown for three different dot radii 15.6 nm, 20.8 nm and 26.0 nm respectively. It is observed that up to a threshold energy linear RIC shows normal dispersion i.e. ∆nr(1)⁄∆nr>0 and above the threshold energy shows anomalous dispersion i.e. ∆nr(1)⁄∆nr<0. The third order nonlinear RIC shows the reverse behavior due to its negative contribution and shows the anomalous dispersion up to the threshold energy and after that shows the normal dispersion. The magnitude of total RIC is reduced before and after the threshold energy due to this negative contribution of third order nonlinear RIC. The RIC is also studied for different value of intensities of incident radiation. A significant change in the magnitude is observed for the total RIC due to change in intensity. The magnitude decreases as the intensity is increased due to increase in nonlinear contribution.
8
Fig. 5. The variation of RIC with incident photon energy
Figure 6 shows the variation of third order nonlinear susceptibility for three different dot radii at three different intensities. It is observed that as the dot radius increases the peak shifts towards lower energy side and on increasing the intensity of the incident radiation the magnitude of third order susceptibility increases.
9
Fig. 6. The variation of third order susceptibility with incident photon energy
CONCLUSIONS In the THz range we have investigated the linear, nonlinear and total photoabsorption coefficient, the RIC and the third order nonlinear susceptibility in for single electron confined in GaAs QD embedded in AlyGa1-yAs matrix. The EMA with finite potential at the dot-matrix interface is used to find energy levels and wavefunctions. It is found that the linear and third order nonlinear photoabsorption coefficient, RIC, and the third order nonlinear susceptibility changes significantly with the variation in intensity of incident radiation and dot radius. We expect that the linear and nonlinear response of intraband transitions in GaAs QDs would be useful in various applications of THz radiation.
ACKNOWLEDGEMENT We would like to thank JIIT, Noida for the facilities and support to carry out this work.
10
REFERENCES [1]
M. Shur, GaAs Devices and Circuits, Springer Science+Business Media, LLC, 1987. doi:10.1007/978-14899-1989-2.
[2]
C.Y. Chang, F. Kai, GaAs High-Speed Devices: Physics, Technology, and Circuit Applications, John Wiley & Sons, New York, 1994.
[3]
P. Harrison, Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures, Second Edition, John Wiley & Sons, 2005.
[4]
U. Banin, Y.W. Cao, D. Katz, O. Millo, Identification of atomic-like electronic states in indium arsenide nanocrystal quantum dots, Nature. 400 (1999) 542–544. doi:10.1038/22979.
[5]
H. Qiang, F.H. Pollak, Y. S. Tang, P.D. Wang, C.M. Sotomayor Torres, Characterization of process‐ induced strains in GaAs/Ga 0.7 Al 0.3 As quantum dots using room‐temperature photoreflectance, Applied Physics Letters. 64 (1994) 2830–2832. doi:10.1063/1.111439.
[6]
T. Guenther, C. Lienau, T. Elsaesser, M. Glanemann, V.M. Axt, T. Kuhn, S. Eshlaghi, A.D. Wieck, Coherent Nonlinear Optical Response of Single Quantum Dots Studied by Ultrafast Near-Field Spectroscopy, Physical Review Letters. 89 (2002) 057401. doi:10.1103/PhysRevLett.89.057401.
[7]
F.G. Tarntair, W.K. Hong, T.K. Ku, N.J. She, C.F. Chen, H.C. Cheng, Fabrication of GaAs quantum dots by modified droplet epitaxy, Japanese Journal of Applied Physics, Part 2: Letters. 39 (2000) L79–L81. doi:10.1143/JJAP.39.L79.
[8]
A. Rastelli, S. Stufler, A. Schliwa, R. Songmuang, C. Manzano, G. Costantini, K. Kern, A. Zrenner, D. Bimberg, O.G. Schmidt, Hierarchical self-assembly of GaAs/AlGaAs quantum dots, Physical Review Letters. 92 (2004) 166104. doi:10.1103/PhysRevLett.92.166104.
[9]
S. Sanguinetti, K. Watanabe, T. Kuroda, F. Minami, Y. Gotoh, N. Koguchi, Effects of post-growth annealing on the optical properties of self-assembled GaAs/AlGaAs quantum dots, Journal of Crystal Growth. 242 (2002) 321–331. doi:10.1016/S0022-0248(02)01434-3.
[10]
K. Brunner, U. Bockelmann, G. Abstreiter, M. Walther, G. Böhm, G. Tränkle, G. Weimann, Photoluminescence from a single GaAs/AlGaAs quantum dot, Physical Review Letters. 69 (1992) 3216– 3219. doi:10.1103/PhysRevLett.69.3216.
[11]
M. Şahin, Third-order nonlinear optical properties of a one- and two-electron spherical quantum dot with and without a hydrogenic impurity, Journal of Applied Physics. 106 (2009) 063710. doi:10.1063/1.3225100.
[12]
B. Çakır, Y. Yakar, A. Özmen, Refractive index changes and absorption coefficients in a spherical quantum dot with parabolic potential, Journal of Luminescence. 132 (2012) 2659–2664. doi:10.1016/j.jlumin.2012.03.065.
[13]
Y. Yakar, B. Çakir, A. Özmen, Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential, Optics Communications. 283 (2010) 1795–1800. doi:10.1016/j.optcom.2009.12.027.
[14]
W. Xie, Linear and nonlinear optical properties of a hydrogenic donor in spherical quantum dots, Physica B: Condensed Matter. 403 (2008) 4319–4322. doi:10.1016/j.physb.2008.09.021.
11
[15]
S. Ünlü, İ. Karabulut, H. Şafak, Linear and nonlinear intersubband optical absorption coefficients and refractive index changes in a quantum box with finite confining potential, Physica E: Low-Dimensional Systems and Nanostructures. 33 (2006) 319–324. doi:10.1016/j.physe.2006.03.163.
[16]
Y. Wang, Nonlinear optical properties of nanometer-sized semiconductor clusters, Accounts of Chemical Research. 24 (1991) 133–139. doi:10.1021/ar00005a002.
[17]
S.P. Purohit, K.C. Mathur, Study of Photoabsorption and Photoelectric Process in GaAs Semiconductor Quantum Dot Nanostructure, Journal of Computational and Theoretical Nanoscience. 7 (2010) 1131–1135. doi:10.1166/jctn.2010.1463.
[18]
T. Brunhes, P. Boucaud, S. Sauvage, A. Lemaître, J.-M. Gérard, F. Glotin, R. Prazeres, J.-M. Ortega, Infrared second-order optical susceptibility in InAs/GaAs self-assembled quantum dots, Physical Review B. 61 (2000) 5562–5570. doi:10.1103/PhysRevB.61.5562.
[19]
S. Yilmaz, M. Şahin, Third-order nonlinear absorption spectra of an impurity in a spherical quantum dot with different confining potential, Physica Status Solidi (B) Basic Research. 247 (2010) 371–374. doi:10.1002/pssb.200945491.
[20]
D. Maikhuri, S.P. Purohit, K.C. Mathur, Quadrupole effects in photoabsorption in ZnO quantum dots, Journal of Applied Physics. 112 (2012). doi:10.1063/1.4767474.
[21]
Anchala, S.P. Purohit, K.C. Mathur, Photoabsorption and photoelectric process in Si nanocrystallites, Applied Physics Letters. 98 (2011) 043106. doi:10.1063/1.3548861.
[22]
D. Krapf, A. Davidi, J. Shappir, A. Sa’ar, Infrared photo-induced absorption spectroscopy of porous silicon, Physica Status Solidi (A). 197 (2003) 566–571. doi:10.1002/pssa.200306564.
[23]
Y.S. Lee, Principles of Terahertz Science and Technology, Springer US, Boston, MA, 2009. doi:10.1007/978-0-387-09540-0.
[24]
M. Tonouchi, Cutting-edge terahertz technology, Nature Photonics. 1 (2007) 97–105. doi:10.1038/nphoton.2007.3.
[25]
A.Y. Pawar, D.D. Sonawane, K.B. Erande, D. V Derle, Terahertz technology and its applications, Drug Invention Today. 5 (2013) 157–163. doi:10.1016/j.dit.2013.03.009.
[26]
E. Pickwell, V.P. Wallace, Biomedical applications of terahertz technology, Journal of Physics D: Applied Physics. 39 (2006) R301–R310. doi:10.1088/0022-3727/39/17/R01.
[27]
J.B. Baxter, G.W. Guglietta, Terahertz Spectroscopy, Analytical Chemistry. 83 (2011) 4342–4368. doi:10.1021/ac200907z.
[28]
M.C. Kemp, P.F. Taday, B.E. Cole, J.A. Cluff, A.J. Fitzgerald, W.R. Tribe, Security applications of terahertz technology, in: R.J. Hwu, D.L. Woolard (Eds.), Proceedings of SPIE - The International Society for Optical Engineering, 2003: p. 44. doi:10.1117/12.500491.
[29]
A.G. Davies, A.D. Burnett, W. Fan, E.H. Linfield, J.E. Cunningham, Terahertz spectroscopy of explosives and drugs, Materials Today. 11 (2008) 18–26. doi:10.1016/S1369-7021(08)70016-6.
[30]
J. Hesler, R. Prasankumar, J. Tignon, Advances in terahertz solid-state physics and devices, Journal of Applied Physics. 126 (2019) 110401. doi:10.1063/1.5122975.
12
[31]
G. Huang, J. Yang, P. Bhattacharya, G. Ariyawansa, A.G.U. Perera, A multicolor quantum dot intersublevel detector with photoresponse in the terahertz range, Applied Physics Letters. 92 (2008) 011117. doi:10.1063/1.2830994.
[32]
X.H. Su, J. Yang, P. Bhattacharya, G. Ariyawansa, A.G.U. Perera, Terahertz detection with tunneling quantum dot intersublevel photodetector, Applied Physics Letters. 89 (2006) 031117. doi:10.1063/1.2233808.
[33]
R. Jacob, S. Winnerl, M. Fehrenbacher, J. Bhattacharyya, H. Schneider, M.T. Wenzel, H. Von Ribbeck, L.M. Eng, P. Atkinson, O.G. Schmidt, M. Helm, Intersublevel Spectroscopy on Single InAs-Quantum Dots by Terahertz Near-Field Microscopy, Nano Letters. 12 (2012) 4336–4340. doi:10.1021/nl302078w.
[34]
E.A. Zibik, T. Grange, B.A. Carpenter, N.E. Porter, R. Ferreira, G. Bastard, D. Stehr, S. Winnerl, M. Helm, H.Y. Liu, M.S. Skolnick, L.R. Wilson, Long lifetimes of quantum-dot intersublevel transitions in the terahertz range, Nature Materials. 8 (2009) 803–807. doi:10.1038/nmat2511.
[35]
R.W. Boyd, Nonlinear Optics, third ed., Elsevier, 2008.
[36]
Anchala, S.P. Purohit, K.C. Mathur, Linear and nonlinear intersubband optical properties of Si quantum dot embedded in oxide, nitride, and carbide matrix, Journal of Applied Physics. 110 (2011) 114320. doi:10.1063/1.3665687.
[37]
D. Maikhuri, S.P. Purohit, K.C. Mathur, Linear and nonlinear intraband optical properties of ZnO quantum dots embedded in SiO2 matrix, AIP Advances. 2 (2012) 012160. doi:10.1063/1.3693405.
[38]
L. Lu, W. Xie, H. Hassanabadi, Linear and nonlinear optical absorption coefficients and refractive index changes in a two-electron quantum dot, Journal of Applied Physics. 109 (2011) 063108. doi:10.1063/1.3560541.
[39]
İ. Karabulut, S. Baskoutas, Linear and nonlinear optical absorption coefficients and refractive index changes in spherical quantum dots: Effects of impurities, electric field, size, and optical intensity, Journal of Applied Physics. 103 (2008) 073512. doi:10.1063/1.2904860.
[40]
C. Delerue, M. Lannoo, Nanostructures: Theory and Modeling, Springer Berlin Heidelberg, 2013.
[41]
S. Adachi, GaAs, AlAs, and AlxGa1− xAs: Material parameters for use in research and device applications, Journal of Applied Physics. 58 (1985) R1–R29. doi:10.1063/1.336070.
[42]
Z.Y. Deng, J. Guo, T. Lai, Impurity states in a spherical GaAs-Ga1−xAlxAs quantum dot: Effects of the spatial variation of dielectric screening, Physical Review B. 50 (1994) 5736–5739. doi:10.1103/PhysRevB.50.5736.
[43]
C. N. Banwell, Fundamentals of Molecular & Spectroscopy, 4th ed., McGraw-Hill Education (India) Pvt Limited, 2001.
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Highlights 1. Terahertz radiation absorption is investigated for single electron GaAs Quantum dot. 2. Third order nonlinear optical properties are studied in terahertz range. 3. Complete photoabsorption spectra is predicted for terahertz region.
S1386-9477(19)31549-8
DOI:
https://doi.org/10.1016/j.physe.2019.113913
Reference:
PHYSE 113913
To appear in:
Physica E: Low-dimensional Systems and Nanostructures
Received Date: 11 October 2019 Accepted Date: 20 December 2019
Please cite this article as: S. Nasa, S.P. Purohit, Linear and third order nonlinear optical properties of GaAs quantum dot in terahertz region, Physica E: Low-dimensional Systems and Nanostructures (2020), doi: https://doi.org/10.1016/j.physe.2019.113913. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
Linear and Third Order Nonlinear Optical Properties of GaAs Quantum Dot in Terahertz Region Sukanya Nasaa) and S. P. Purohit Department of Physics and Materials Science and Engineering Jaypee Institute of Information Technology, A-10, Sector-62, Noida, 201309, India
a)
Corresponding author: [email protected]
Abstract: We study the electronic and optical properties of GaAs quantum dot embedded in Ga1-yAlyAs matrix. Using the effective mass approximation with the finite confinement potential the intraband energy levels and the wavefunctions are obtained. The linear and third order nonlinear optical properties, photoabsorption coefficient, refractive index change and susceptibility are studied for different dot radius in the terahertz region.
Keywords: Quantum dots, Third order nonlinearity, Terahertz Radiation, Nanomaterials
INTRODUCTION The physical, optical and electronic properties of GaAs have made it an important compound for various technological applications[1],[2]. The direct band gap electronic structure and the noncentrosymmetric nature of GaAs are advantageous for its optoelectronic applications. The electronic and optical properties changes to a great extent in quantum wells, quantum wires and quantum dots (QDs)[3] due to quantum confinement effects. In QD structures, the electronic properties are very similar to that of atoms. The QDs are also called artificial atoms[4]. In QDs energy levels are configurable by changing only few physical parameters like radius of the dot, size and/or shape of the dot or the change of surrounding matrix material[3]. Different experimental techniques [5], [6], [7] are reported in the literature to fabricate and study the properties of GaAs QDs. Rastelli et al.[8]
fabricated self-assembled unstrained GaAs QDs of tunable sizes by a suitable
combination of Stranski-Krastanow (SK) growth and etching. Sanguinetti et al.[9] studied the optical properties of self-assembled GaAs/GaAlAs QDs grown by modified droplet expitaxy method. Brunner et al.[10] fabricated single QDs of different sizes (250-2000 nm) by laser induced local inter-diffusion of GaAs/GaAlAs quantum wells. In recent past, theoretical studies on the linear optical properties (LOP) and nonlinear optical properties (NLOP) of GaAs nano structures have gained a considerable interest[11], [12], [13], [14],[15], [16]. Using full numeric matrix
1
diagonalization method Sahin[11] studied the third order nonlinear photoabsorption for single and double electron QDs with and without hydrogenic impurity. The Quantum Genetic Algorithm (QGA) and Hartree–Fock–Roothaan (HFR) with parabolic confinement potential are used by Cakir et al.[12] to obtain wavefunctions and energy Eigen values of different confined states to study the refractive index changes (RIC) and absorption coefficients in spherical QDs. Using the effective mass approximation (EMA) with the parabolic confinement Purohit and Mathur[17] studied the photoabsorption and photoelectric processes in GaAs QD structures. In this study we investigate the LOP and third order NLOP in terahertz region for single electron confined in spherical QD. The wave functions and energies of confined states of the conduction band are obtained by using the EMA with finite potential. The conduction band offset at the dot-matrix interface is considered as the confinement potential. Some studies on intraband optical properties of different QDs are reported in literature. Using the EMA[18][19][20] with the finite potential Anchala et al.[21] investigated the photoabsorption and photoelectric process in one electron QD and found a good agreement with experimental data[22]. “Terahertz (THz) radiation” is a term usually given to electromagnetic radiation of frequency range from 0.1 THz to 10 THz[23][24][25]. Due to its peculiar properties such as non-ionizing and non-damaging characteristics, THz covers a wide range of applications such as biomedical imaging[26], ultrafast spectroscopy[27], security[28], material identification[29] and several others[24][30]. The devices based on THz radiations are in general fast and sensitive as compared to the devices based on other regions of electromagnetic spectrum. Despite their applications in broad areas, until recently THz radiations were mostly used for spectroscopic applications. One of the main causes of its limited applicability was due to unavailability of THz sources and detectors. Recent advances in nanotechnology and optoelectronics have resulted in possibility of nano structure based THz sources and detectors. Some studies are reported in the literature for THz generation and detection from InAs and InGaAs QD structures[31][32][33][34]. The experimental and theoretical investigations are timely in demand to explore the THz sources and detectors based on GaAs nanostructures. In this study we investigate the LOP and third order NLOP like photoabsorption, RIC and susceptibility for a range of dot size in THz region.
THEORY The nonlinear optical response in terms of polarization due to incident radiation of field strength is often
described as[35],
= [ + + + ⋯ ]
where, is the n order nonlinear susceptibility and is permittivity of the free space. For a two level system
(1)
th
and are defined as [11][15][36][37] ,
2 ℇ − ℇ − ℏ − ℏγ
= and
= −
# | | 4 ' − ( % − ) 3 ℇ − ℇ − ℏ − ℏγ ℇ − ℇ − ℏ + ℏγ 'ℇ − ℇ − ℏγ('ℇ − ℇ − ℏ − ℏγ(
2
where, is carrier density in QD, ℇ is the energy of electronic state n. refers to electronic charge and f and i stands for final and initial states respectively and Tfi is the transition matrix for a transition i→f. ℏ is the energy of
incident radiation and γ is line width.
The optical absorption coefficient and the change in refractive index are the imaginary and real part of the susceptibility.
The optical absorption coefficient + is given as[36][38][39] ,
+ = ,
/0[ ] 4 .
where, - and . are permeability and real part of permittivity of the QD respectively. The linear and third order
nonlinear absorption coefficient are given by, + =
1 5'ℇ − ℇ − ℏ( 5 23 4
and + = −
− 2 # 1 / # 5 'ℇ − ℇ − ℏ( 71 − 23 4 ℏγ 4 ×
: ℇ − ℇ − ℏ − ℏγ + 2'ℇ − ℇ ('ℇ − ℇ − ℏ(; < 6
ℇ − ℇ + ℏγ
where, 5 is the delta function, 23 is the refractive index of the dot material, c is speed of light and I is intensity of incident radiation.
The 5 function is defined as
5=
ℏγ
1: ℇ − ℇ − ℏ + ℏγ ;
7
The total photoabsorption coefficient + ? is defined as,
+ ? = + + + 8 The susceptibility is related to the RIC by
∆2 = B C D 9 23 223 The corresponding linear and third order nonlinear RIC are given by
ℇ − ℇ − ℏ ∆2 = C D 10 23 2 23
ℇ − ℇ − ℏ + ℏγ
and
' − ( ∆2 # -4/ = − G4'ℇ − ℇ − ℏ( − 23 4 23 : ℇ − ℇ − ℏ + ℏγ ;
ℇ − ℇ + ℏγ × H'ℇ − ℇ − ℏ( × :'ℇ − ℇ ( × 'ℇ − ℇ − ℏ( − ℏγ ;
− ℏγ '2'ℇ − ℇ ( − ℏ(IJ 11
3
The total RIC
∆2 ? K 23 is defined as
∆2 ? ∆2 ∆2 = + 12 23 23 23 The transition matrix element is defined as,
= ℱMΨ O. ̂Ψ R 13
where, ℱ is local field factor which relates the electric field inside and outside of the dot[40]. ̂ is unit polarization
vector of the linearly polarized light.
Using the EMA, wavefunctions of the QD states ΨST O, θ, W = BS O YST θ, φ are obtained by solving
Schrodinger equation, Z−
ℏ ∇ + ]^ O _ ΨST O, θ, W = ΨST O, θ, W 14 20∗
where, ]^ O is the finite potential at the interface of the dot and matrix. BS O and YST θ, φ are radial and
angular parts of the wave function respectively and n, l and m are quantum numbers. The radial part of the wave function is considered in terms of the spherical Bessel (`S ) and modified spherical Bessel function (aS ) as, B,S = b
`S cO deO O < B gaS hO deO O > B
(15)
where, P and Q are the normalization constant. R is the dot radius and parameters ζ and η are given by[40] c=, and
∗ 20 ℇS 16 ℏ
h = ,j k
∗ 2]^ 0 − c l 17 ℏ
∗ ∗ where, 0 is the effective mass of electron inside the dot and 0mno is the effective mass of electron in the matrix
material. ρ is the ratio
∗ 0 p0∗ . Applying the appropriate boundary conditions the following transcendental mno
equation is obtained which is solved numerically jc
aSq hB r + 1 `Sq cB
j − 1 18 = −h + `S cB aS hB B The effective finite potential ]^ is given by,
]^ = b
0 , deO O < B 19 ] + ]s , deO O ≥ B
where, V0 is the conduction band offset at the dot matrix interface. Vs is the self-energy given by[40]
1 1 1 ]s = C − D + ∆]s 20 2 u3 mno u3 41u B
where,
4
∆]s ≈ 0.466
u3 − u3 mno C D 21 41u u3 B u3 − u3 mno
We use the conduction band offset and dielectric constant for the mole fraction of aluminium y=0.3 as reported by Adachi[41][42]. The values of different parameters used in the present study are given in Table 1. The atomic units are used throughout. Table 1. Values of parameters used in calculations
Parameter u3
u3 mno =13.18-3.12y
Value 13.18 12.244
∗ 0
0.067 mx
V =0.6(1.155y+0.37y2) eV
0.277 eV
Effective Bohr radius a0*
10.4 nm
Effective Hartree energy Ht*
10.5 meV
∗ 0mno =(0.067+0.083y)me
0.0919 mx
RESULTS AND DISCUSSIONS In this study we investigate the LOP and NLOP in THz region associated with intraband transitions in singly charged spherical QD of GaAs embedded in Ga1-yAlyAs. Figure 1 shows the radial part of the wavefunctions for 1s, 1p, 2s and 2p discrete energy states of QD of radius 15.6 nm.
Fig. 1. Radial wavefunctions of some energy states
5
Figure 2(a) shows various discrete energy levels of l=0, 1, 2 and 3 in the conduction band of the dot radius of 15.6 nm. Figure 2(b) shows the variation of transition energies of (1s-1p), (1p-1d), (1d-1f), (1p-2s), (2s-2p) and (2p2d) dipole transitions for the dot radius range from 10.4 nm to 30 nm. The transition energy decreases on increasing the dot radius. The transition energy in THz range is marked for the dot radius range from 10.4 nm to 30 nm.
Fig. 2(a). Discrete energy levels in the dot of radius 15.6 nm
Fig. 2(b). Variation of transition energies with dot radius
To explore the effect of changing the molar fraction of Al in the surrounding matrix we also estimate the (1s-1p) transition energy for three values of y, 0.2, 0.3 and 0.4 at dot radius R=15.6 nm. The (1s-1p) transition energy is obtained as 18.215 meV, 19.049 and 20.303 meV for the molar fraction of 0.2, 0.3 and 0.4 respectively. The change in the transition energy is only up to about 2 meV.
Figure 3 shows the total photoabsorption spectra + ? (Eqn. 8) consisting of all dipole transitions for QDs of
radius 15.6 nm and at intensity of incident radiation I= 20 MW/m2. To predict the total photoabsorption spectra in experimental
situation
with
an
assembly
of
QDs
the
effect
of
ratio
of
probability[43]
ℇ − ℇ s K z Kz = {| }− ~ a of QDs in their initial state i at temperature T with respect to the ground state 1s s
6
is included. a is the Boltzmann constant. Factor of the ratio of probability is taken at room temperature T=300K.
The photoabsorption peaks are obtained up to the incident photon energy of 220 meV. It is found that the absorption peaks in THz region are obtained at 19 meV, 23.9 meV, 28.5 meV, 35.5 meV, 36.8 meV and 41.3 meV
corresponding to (1s-1p), (1p-1d), (1d-1f), (1p-2s), (2s-2p) and (2p-2d) transitions respectively. Transitions corresponding to all absorption peaks are marked in the figure. The overlapping of absorption peaks corresponding to (1p-3s) and (2s-2p); (2d-2f) and (1d-2p); and (1f-2d) and (2d-3p) transitions is due to their approximately equal transition energies. In absorption spectra peaks corresponding to all dipole transitions are obtained. The height of different absorption peaks is found changing maximum for the 1s-1p transition to minimum for the 1d-3p transition. This is due to the different value of the transition matrix element for different transitions and the ratio of probability of the initial state. The highest absorption peak in the THz range corresponds to the 1s-1p transition and we further investigate the LOP and third order NLOP viz; the photoabsorption, RIC and the susceptibility in THz region. We investigate the effect of change in dot size and change in intensity of incident radiation on the NLOP.
Fig. 3. Variation of total photoabsorption coefficient with incident photon energy
Figure 4 shows the change in the linear absorption, the nonlinear absorption and the total absorption coefficients with the variation of incident photon energy at three different dot radii, 15.6 nm, 20.8 nm and 26 nm respectively with three values of intensities 20 MW/m2, 30 MW/m2 and 40 MW/m2. The photoabsorption peaks are observed at
7
19 meV, 11.5 meV and 7.5 meV for the dot radius R 15.6 nm, 20.8 nm and 26 nm respectively. It is observed that on increasing the dot radius (Fig. 3 a, b and c) photoabsorption peak shifts towards lower energy values. This can be understood by considering the dependence of photoabsorption coefficient in inversely proportional manner to the volume of dot. At a fixed dot radius and increasing the intensity of the incident radiation (Fig. 3 a, d and g), the peak height decreases. This is due to the negative nonlinear contribution, which increases with increasing intensity.
Fig. 4. The variation of photoabsorption coefficient with incident photon energy
Figure 5 shows the RIC with the variation of incident photon energy. Results are shown for three different dot radii 15.6 nm, 20.8 nm and 26.0 nm respectively. It is observed that up to a threshold energy linear RIC shows normal dispersion i.e. ∆nr(1)⁄∆nr>0 and above the threshold energy shows anomalous dispersion i.e. ∆nr(1)⁄∆nr<0. The third order nonlinear RIC shows the reverse behavior due to its negative contribution and shows the anomalous dispersion up to the threshold energy and after that shows the normal dispersion. The magnitude of total RIC is reduced before and after the threshold energy due to this negative contribution of third order nonlinear RIC. The RIC is also studied for different value of intensities of incident radiation. A significant change in the magnitude is observed for the total RIC due to change in intensity. The magnitude decreases as the intensity is increased due to increase in nonlinear contribution.
8
Fig. 5. The variation of RIC with incident photon energy
Figure 6 shows the variation of third order nonlinear susceptibility for three different dot radii at three different intensities. It is observed that as the dot radius increases the peak shifts towards lower energy side and on increasing the intensity of the incident radiation the magnitude of third order susceptibility increases.
9
Fig. 6. The variation of third order susceptibility with incident photon energy
CONCLUSIONS In the THz range we have investigated the linear, nonlinear and total photoabsorption coefficient, the RIC and the third order nonlinear susceptibility in for single electron confined in GaAs QD embedded in AlyGa1-yAs matrix. The EMA with finite potential at the dot-matrix interface is used to find energy levels and wavefunctions. It is found that the linear and third order nonlinear photoabsorption coefficient, RIC, and the third order nonlinear susceptibility changes significantly with the variation in intensity of incident radiation and dot radius. We expect that the linear and nonlinear response of intraband transitions in GaAs QDs would be useful in various applications of THz radiation.
ACKNOWLEDGEMENT We would like to thank JIIT, Noida for the facilities and support to carry out this work.
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REFERENCES [1]
M. Shur, GaAs Devices and Circuits, Springer Science+Business Media, LLC, 1987. doi:10.1007/978-14899-1989-2.
[2]
C.Y. Chang, F. Kai, GaAs High-Speed Devices: Physics, Technology, and Circuit Applications, John Wiley & Sons, New York, 1994.
[3]
P. Harrison, Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures, Second Edition, John Wiley & Sons, 2005.
[4]
U. Banin, Y.W. Cao, D. Katz, O. Millo, Identification of atomic-like electronic states in indium arsenide nanocrystal quantum dots, Nature. 400 (1999) 542–544. doi:10.1038/22979.
[5]
H. Qiang, F.H. Pollak, Y. S. Tang, P.D. Wang, C.M. Sotomayor Torres, Characterization of process‐ induced strains in GaAs/Ga 0.7 Al 0.3 As quantum dots using room‐temperature photoreflectance, Applied Physics Letters. 64 (1994) 2830–2832. doi:10.1063/1.111439.
[6]
T. Guenther, C. Lienau, T. Elsaesser, M. Glanemann, V.M. Axt, T. Kuhn, S. Eshlaghi, A.D. Wieck, Coherent Nonlinear Optical Response of Single Quantum Dots Studied by Ultrafast Near-Field Spectroscopy, Physical Review Letters. 89 (2002) 057401. doi:10.1103/PhysRevLett.89.057401.
[7]
F.G. Tarntair, W.K. Hong, T.K. Ku, N.J. She, C.F. Chen, H.C. Cheng, Fabrication of GaAs quantum dots by modified droplet epitaxy, Japanese Journal of Applied Physics, Part 2: Letters. 39 (2000) L79–L81. doi:10.1143/JJAP.39.L79.
[8]
A. Rastelli, S. Stufler, A. Schliwa, R. Songmuang, C. Manzano, G. Costantini, K. Kern, A. Zrenner, D. Bimberg, O.G. Schmidt, Hierarchical self-assembly of GaAs/AlGaAs quantum dots, Physical Review Letters. 92 (2004) 166104. doi:10.1103/PhysRevLett.92.166104.
[9]
S. Sanguinetti, K. Watanabe, T. Kuroda, F. Minami, Y. Gotoh, N. Koguchi, Effects of post-growth annealing on the optical properties of self-assembled GaAs/AlGaAs quantum dots, Journal of Crystal Growth. 242 (2002) 321–331. doi:10.1016/S0022-0248(02)01434-3.
[10]
K. Brunner, U. Bockelmann, G. Abstreiter, M. Walther, G. Böhm, G. Tränkle, G. Weimann, Photoluminescence from a single GaAs/AlGaAs quantum dot, Physical Review Letters. 69 (1992) 3216– 3219. doi:10.1103/PhysRevLett.69.3216.
[11]
M. Şahin, Third-order nonlinear optical properties of a one- and two-electron spherical quantum dot with and without a hydrogenic impurity, Journal of Applied Physics. 106 (2009) 063710. doi:10.1063/1.3225100.
[12]
B. Çakır, Y. Yakar, A. Özmen, Refractive index changes and absorption coefficients in a spherical quantum dot with parabolic potential, Journal of Luminescence. 132 (2012) 2659–2664. doi:10.1016/j.jlumin.2012.03.065.
[13]
Y. Yakar, B. Çakir, A. Özmen, Calculation of linear and nonlinear optical absorption coefficients of a spherical quantum dot with parabolic potential, Optics Communications. 283 (2010) 1795–1800. doi:10.1016/j.optcom.2009.12.027.
[14]
W. Xie, Linear and nonlinear optical properties of a hydrogenic donor in spherical quantum dots, Physica B: Condensed Matter. 403 (2008) 4319–4322. doi:10.1016/j.physb.2008.09.021.
11
[15]
S. Ünlü, İ. Karabulut, H. Şafak, Linear and nonlinear intersubband optical absorption coefficients and refractive index changes in a quantum box with finite confining potential, Physica E: Low-Dimensional Systems and Nanostructures. 33 (2006) 319–324. doi:10.1016/j.physe.2006.03.163.
[16]
Y. Wang, Nonlinear optical properties of nanometer-sized semiconductor clusters, Accounts of Chemical Research. 24 (1991) 133–139. doi:10.1021/ar00005a002.
[17]
S.P. Purohit, K.C. Mathur, Study of Photoabsorption and Photoelectric Process in GaAs Semiconductor Quantum Dot Nanostructure, Journal of Computational and Theoretical Nanoscience. 7 (2010) 1131–1135. doi:10.1166/jctn.2010.1463.
[18]
T. Brunhes, P. Boucaud, S. Sauvage, A. Lemaître, J.-M. Gérard, F. Glotin, R. Prazeres, J.-M. Ortega, Infrared second-order optical susceptibility in InAs/GaAs self-assembled quantum dots, Physical Review B. 61 (2000) 5562–5570. doi:10.1103/PhysRevB.61.5562.
[19]
S. Yilmaz, M. Şahin, Third-order nonlinear absorption spectra of an impurity in a spherical quantum dot with different confining potential, Physica Status Solidi (B) Basic Research. 247 (2010) 371–374. doi:10.1002/pssb.200945491.
[20]
D. Maikhuri, S.P. Purohit, K.C. Mathur, Quadrupole effects in photoabsorption in ZnO quantum dots, Journal of Applied Physics. 112 (2012). doi:10.1063/1.4767474.
[21]
Anchala, S.P. Purohit, K.C. Mathur, Photoabsorption and photoelectric process in Si nanocrystallites, Applied Physics Letters. 98 (2011) 043106. doi:10.1063/1.3548861.
[22]
D. Krapf, A. Davidi, J. Shappir, A. Sa’ar, Infrared photo-induced absorption spectroscopy of porous silicon, Physica Status Solidi (A). 197 (2003) 566–571. doi:10.1002/pssa.200306564.
[23]
Y.S. Lee, Principles of Terahertz Science and Technology, Springer US, Boston, MA, 2009. doi:10.1007/978-0-387-09540-0.
[24]
M. Tonouchi, Cutting-edge terahertz technology, Nature Photonics. 1 (2007) 97–105. doi:10.1038/nphoton.2007.3.
[25]
A.Y. Pawar, D.D. Sonawane, K.B. Erande, D. V Derle, Terahertz technology and its applications, Drug Invention Today. 5 (2013) 157–163. doi:10.1016/j.dit.2013.03.009.
[26]
E. Pickwell, V.P. Wallace, Biomedical applications of terahertz technology, Journal of Physics D: Applied Physics. 39 (2006) R301–R310. doi:10.1088/0022-3727/39/17/R01.
[27]
J.B. Baxter, G.W. Guglietta, Terahertz Spectroscopy, Analytical Chemistry. 83 (2011) 4342–4368. doi:10.1021/ac200907z.
[28]
M.C. Kemp, P.F. Taday, B.E. Cole, J.A. Cluff, A.J. Fitzgerald, W.R. Tribe, Security applications of terahertz technology, in: R.J. Hwu, D.L. Woolard (Eds.), Proceedings of SPIE - The International Society for Optical Engineering, 2003: p. 44. doi:10.1117/12.500491.
[29]
A.G. Davies, A.D. Burnett, W. Fan, E.H. Linfield, J.E. Cunningham, Terahertz spectroscopy of explosives and drugs, Materials Today. 11 (2008) 18–26. doi:10.1016/S1369-7021(08)70016-6.
[30]
J. Hesler, R. Prasankumar, J. Tignon, Advances in terahertz solid-state physics and devices, Journal of Applied Physics. 126 (2019) 110401. doi:10.1063/1.5122975.
12
[31]
G. Huang, J. Yang, P. Bhattacharya, G. Ariyawansa, A.G.U. Perera, A multicolor quantum dot intersublevel detector with photoresponse in the terahertz range, Applied Physics Letters. 92 (2008) 011117. doi:10.1063/1.2830994.
[32]
X.H. Su, J. Yang, P. Bhattacharya, G. Ariyawansa, A.G.U. Perera, Terahertz detection with tunneling quantum dot intersublevel photodetector, Applied Physics Letters. 89 (2006) 031117. doi:10.1063/1.2233808.
[33]
R. Jacob, S. Winnerl, M. Fehrenbacher, J. Bhattacharyya, H. Schneider, M.T. Wenzel, H. Von Ribbeck, L.M. Eng, P. Atkinson, O.G. Schmidt, M. Helm, Intersublevel Spectroscopy on Single InAs-Quantum Dots by Terahertz Near-Field Microscopy, Nano Letters. 12 (2012) 4336–4340. doi:10.1021/nl302078w.
[34]
E.A. Zibik, T. Grange, B.A. Carpenter, N.E. Porter, R. Ferreira, G. Bastard, D. Stehr, S. Winnerl, M. Helm, H.Y. Liu, M.S. Skolnick, L.R. Wilson, Long lifetimes of quantum-dot intersublevel transitions in the terahertz range, Nature Materials. 8 (2009) 803–807. doi:10.1038/nmat2511.
[35]
R.W. Boyd, Nonlinear Optics, third ed., Elsevier, 2008.
[36]
Anchala, S.P. Purohit, K.C. Mathur, Linear and nonlinear intersubband optical properties of Si quantum dot embedded in oxide, nitride, and carbide matrix, Journal of Applied Physics. 110 (2011) 114320. doi:10.1063/1.3665687.
[37]
D. Maikhuri, S.P. Purohit, K.C. Mathur, Linear and nonlinear intraband optical properties of ZnO quantum dots embedded in SiO2 matrix, AIP Advances. 2 (2012) 012160. doi:10.1063/1.3693405.
[38]
L. Lu, W. Xie, H. Hassanabadi, Linear and nonlinear optical absorption coefficients and refractive index changes in a two-electron quantum dot, Journal of Applied Physics. 109 (2011) 063108. doi:10.1063/1.3560541.
[39]
İ. Karabulut, S. Baskoutas, Linear and nonlinear optical absorption coefficients and refractive index changes in spherical quantum dots: Effects of impurities, electric field, size, and optical intensity, Journal of Applied Physics. 103 (2008) 073512. doi:10.1063/1.2904860.
[40]
C. Delerue, M. Lannoo, Nanostructures: Theory and Modeling, Springer Berlin Heidelberg, 2013.
[41]
S. Adachi, GaAs, AlAs, and AlxGa1− xAs: Material parameters for use in research and device applications, Journal of Applied Physics. 58 (1985) R1–R29. doi:10.1063/1.336070.
[42]
Z.Y. Deng, J. Guo, T. Lai, Impurity states in a spherical GaAs-Ga1−xAlxAs quantum dot: Effects of the spatial variation of dielectric screening, Physical Review B. 50 (1994) 5736–5739. doi:10.1103/PhysRevB.50.5736.
[43]
C. N. Banwell, Fundamentals of Molecular & Spectroscopy, 4th ed., McGraw-Hill Education (India) Pvt Limited, 2001.
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Highlights 1. Terahertz radiation absorption is investigated for single electron GaAs Quantum dot. 2. Third order nonlinear optical properties are studied in terahertz range. 3. Complete photoabsorption spectra is predicted for terahertz region.