Institute of Applied Problems of Physics: Research on Quantum Dots and Ellipsoidal Structures
Explore the intriguing study conducted by the Institute of Applied Problems of Physics (IAPP) on quantum dots and ellipsoidal structures, focusing on few-particle intraband dipole transitions in strongly oblate asymmetric ellipsoid QDs. Dive into the topics of ion channeling, Moshinsky model, Kohn's theorem, and the effects of external electric fields on these nanostructures. Discover the detailed analysis and results of channeling through quantum dots and the investigation of strains in synthesized QDs. Explore the complexities of the one-particle problem in these innovative quantum systems.
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Institute of Applied Problems of Physics (IAPP) Few-particle intraband dipole transitions in strongly oblate asymmetric ellipsoid QD Yevgeni Mamasakhlisov In collaboration with Aram Nahapetyan, Mher Mkrtchyan, Hayk Sarkisyan Riccione 2024
Outline: Ion channeling and QDs QD and one particle problem Few particle problem and Moshinsky model Kohn s theorem and QD The effect of external electric field Results
Channeling through QDs Atomic force microscope (AFM) photograph of the uncapped sample Schematic view of (a) the uncapped and (b) the capped QDs samples RBS/w channeling measurement was performed with 4 MeV 12C2+ beam provided by the 9SDH-2 Tandem accelerator. The obtained results show the existence of strains in and around the synthesized QDs.
Ellipsoidal Quantum Dot Z O Y X
One particle problem = = ( , , ) x y z ( , , ) x y z ( ;( , )) z x y ( ;( , )) z x y ( , ) x y ( , ) x y f f s s 2 2 n n = = + + ( ;( , )) z x y ( ;( , )) z x y sin sin , , z z z z f f n n ( , ) L x y ( , ) L x y ( , ) L x y ( , ) L x y z z 2 2 2 2 2 2 z z n n = = z z n n ( , ) x y ( , ) x y E E ( , ) L x y ( , ) L x y 2 2 2 2 z z 2 2 2 2 x y x a a y b b = = ( , ) L x y ( , ) L x y 2 2 1 1 c c 2 2 2 2
One particle problem 1 1 1 x 1 x a a = = 2 2 2 2 2 2 ( , ) L x y ( , ) L x y y y b b 2 2 4 4 1 1 c c 2 2 2 2 x a x y y a b b 2 2 2 2 1 1 1 x y 1 c c x a a y b b 2 2 2 2 2 2 2 2 2 2 2 2 a a 2 b b 2 z z n x ( ) n x n y ( ) n y n n ( ) 2 2 ( ) 2 2 + + + + = = + + + + 1 1 z z n n ( , ) x y ( , ) x y E E z z z z 2 2 2 2 2 2 2 2 2 2 ( , ) L x y ( , ) L x y 4 4 8 8 c c z z 2 n ac n bc bc n ac n = = ( ) n ( ) n z z a a z z 2 2 = = ( ) n ( ) n z z b b z z 2
One particle problem: 2D Schrodinger equation 2 2 2 2 2 2 2 a a 2 b b 2 2 2 2 2 2 n x ( ) n x n y ( ) n y ( ) 2 2 ( ) 2 2 + + + + + + = = ( , ) x y ( , ) x y ( , ) x y ( , ) x y ( , ) x y ( , ) x y z z z z s s s s s s 2 2 2 2 2 2 x x y y 2 2 2 2 2 2 z z n n = = , , x y x y n n n n ( , ) x y ( , ) x y ( , ). x y ( , ). x y E E E E , , s s s s 2 2 8 8 c c x x y y
One particle Wave function and Energy Spectrum 2 2 2 2 x x y y 2 2 n n x x y y 2 2 2 2 2 2 2 2 a a a a = = + + ( , , ) x y z ( , , ) x y z sin sin z z C e H H C e C e H H z z C e ( ( ) ) ( ( ) ) n n n n a a z z a a z z n n n n n n n n n n ( , ) L x y ( , ) L x y ( , ) L x y ( , ) L x y a a a a z z x x x x y y y y ( ( ) ) ( ( ) ) n n n n a a z z a a z z 2 2 2 2 2 2 z z 1 1 1 2 2 1 2 2 n n = = + + + + + + + + ( ) n ( ) n ( ) n ( ) n n n n E n n n E n n n n , , , , a a z z x x b b z z y y 2 2 8 8 c c x x y y z z
Few particle problem N N = = F F ( ,..., r ( ,..., r ) ) ( ) f z ( ) f z ( ( ,..., ,..., ) ) r r 1 1 N N 1 1 N N = = 1 1 i i + + 2 2 2 x x 2 y y + + 2 2 2 2 z z 2 a a 2 i i 2 b b 2 i i z i i z i i n x ( ) n x n y ( ) n y 1 N N N N N N N N N N P P P P ( ) 2 2 ( ) 2 2 1 2 2 + + + + = = ( , , i v x y x y ( , , i i v x y x y , , ) ) ( , ,..., x y ( , ,..., x y , , ) ) x x y y i i i i 1 1 1 1 j j j j s s N N N N i 2 2 = = = = = = = = i j j i j j = = i j i j 1 1 1 1 1 1 1, 1, 1, 1, i i i i i i i i 2 2 2 2 2 2 z z i i n N N n c c = = ( , ,..., x y ( , ,..., x y , , ) ) E E x x y y 1 1 1 1 s s N N N N 2 2 8 8 = = 1 1 i i
Moshinsky Model 1 1 N N N N N N N N N N N N 1 2 2 1 2 2 = = + + 2 2 2 2 ( , v x y x y ( , i v x y x y , , , , ) ) ( ( ) ) ( ( ) ) x x x x y y y y 1 1 2 2 i i j j j j i i j j i i j j i = = i j j i j j = = i j i j = = i j j i j j = = i j i j = = i j j i j j = = i j i j 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, i i i i i i 2 = = = = = = = = = = z z z z z z N N ( ( ) ) ( ( ) ) ... ... ( ( ) ) (1) (1) n n n n n n 1 1 2 2 a a a a a a a a ac ac 2 2 = = = = = = = = = = z z z z z z N N ( ( ) ) ( ( ) ) ... ... ( ( ) ) (1) (1) n n n n n n 1 1 2 2 b b b b b b b b bc bc 2 + + 2 2 2 x x 2 y y + + 2 2 2 a a 2 b b (1) (1) (1) (1) 1 1 N N N N N N N N N N N N N N P P P P 1 2 2 1 2 2 + + + + + + 2 2 2 2 2 2 2 i i 2 i i ( ( ) ) ( ( ) ) x x y y x x x x y y y y i i i i 1 1 2 2 i i j j i i j j 2 2 2 2 2 2 = = = = = = = = i j j i j j = = i j i j = = i j j i j j = = i j i j 1 1 1 1 1 1 1, 1, 1, 1, 1, 1, 1, 1, i i i i i i i i i i 2 2 2 2 = = ( ,..., x ( ,..., x , , ,..., ,..., ) ) ( ,..., x ( ,..., x , , ,..., ,..., ) ) ( ) N ( ) N ( ,..., x ( ,..., x , , ,..., ,..., ) ) x x y y y y E E N N x x y y y y x x y y y y 1 1 1 1 1 1 1 1 1 1 1 1 s s N N N N s s N N N N s s N N N N 2 2 8 8 c c
Two Schrodinger equations along the axes OX and OY 2 2 a a (1) (1) 1 1 N N N N N N N N 1 2 1 2 2 2 2 x x + + + + = = x 2 2 2 x x x 2 i i x s s x s s ( ( ) ) ( ,..., x ( ,..., x ) ) ( ,..., x ( ,..., x ) ) P P x x x x x x x x E E x x 1 1 1 1 1 1 i i j j N N N N i i 2 2 2 = = = = = = i j j i j j = = i j i j 1 1 1 1 1, 1, 1, 1, i i i i i i 2 2 b b (1) (1) 1 1 1 2 1 2 2 N N N N N N N N 2 2 y y + + + + = = y 2 2 2 y y y 2 i i y s s y s s ( ( ) ) ( ,..., y ( ,..., y ) ) ( ,..., y ( ,..., y ) ) P P y y y y y y y y E E y y 2 2 1 1 1 1 i i j j N N N N i i 2 2 2 = = = = = = i j j i j j = = i j i j 1 1 1 1 1, 1, 1, 1, i i i i i i x x y y ( ) N ( ) N E E E E
Transition to new variables and dimensionless Schrodinger equations for X and Y y y x x y y E E x x E E = = = = = = = = = = = = , , ( ) ( ) , , W N W N g g 2 2 , , ( ) ( ) , , W N W N g g 1 2 2 2 b b 1 2 a a 2 2 y y 1 1 x x (1) (1) (1) (1) (1) (1) (1) (1) b b a a (1) (1) (1) (1) b b a a 2 2 x x s s 1 1 N N N N N N N N g g 1 2 2 1 2 2 i + + + + = = x x 2 2 2 x x 2 i i x s s x s s ( ( ) ) 1 1 W W i i j j 2 2 i i 2 2 = = = = = = = = 1 1 1 1 1 1 1 1 i i i i j j i i j i j 2 2 y s y s 2 1 1 N N N N N N N N g g 1 2 2 1 2 2 i + + + + = = y y 2 2 2 y y 2 i i y s s y s s ( ( ) ) 2 2 W W i i j j 2 i i 2 2 = = = = = = = = 1 1 1 1 1 1 1 1 i i i i j j i i j i j
Jacobi coordinates and Schrodinger equations + + + + ... 1 1 1 1 1 i i i i ... N N 1 = = = = = = , , , , 2,3,..., 2,3,..., 1 1 2 2 N N X X X X i i N N i i i i k k 1 1 i i i i = = 1 1 k k + + + + ... 1 1 1 1 1 i i ... N N i i 1 = = = = = = , , , , 2,3,..., 2,3,..., 1 1 2 2 N N Y Y Y Y i i N N i i i i k k 1 1 i i i i = = 1 1 k k 2 2 2 2 2 2 1 1 1 1 2 2 1 2 2 1 2 2 N N X X 2 2 a a + + + + + + = = x x 2 2 i i x x x s s x s s , , X X W W = = 1 2 + 1 2 + Ng Ng a a 2 2 2 2 i i 2 2 X X X X 1 1 = = 2 2 i i = = 1 2 + 1 2 + Ng Ng b b 2 2 2 2 2 2 2 2 1 1 1 N N 1 2 2 1 2 2 1 2 2 Y Y 2 2 b b + + + + + + = = y y 2 2 y y y s s y s s Y Y W W i i 2 2 2 2 2 2 Y Y Y Y = = 2 2 i i i i
System energy spectrum and wave functions 2 2 2 2 1 1 1 1 N N N N 1 2 2 1 2 2 1 2 2 1 2 2 i = = + + + + + + + + + + + + + + + + x x x y y y x rel i n rel i n y rel i n rel i n (1) (1) (1) (1) (1) (1) (1) (1) E E N N . . c m n . . c m n . . c m n . . c m n a a b b . . . state state a a a a b b b b . 2 2 8 8 c c = = = = 2 2 2 2 i i i x x x y y y x rel rel y rel rel , , , , , , state state . . c m n . . c m n n n . . c m n . . c m n n n 1/4 ) ( 1/4 ) ( 1/4 1/4 N N 1 1 1 1 1 1 a a 2 2 i i X X ( ( ) ) a a 2 2 = = /2 /2 X X , , ( ) X ( ) X , , X X X X e e H H e e H H X X 2 2 a a i i i i x x x x reli reli . . c m n . . c m n n n x x x x reli reli . . c m n . . c m n n n x x 2 2 ! ! . . c m n . . c m n x x rel rel 2 2 ! ! n n = = 2 2 i i i i 1/4 ) ( 1/4 ) ( 1/4 1/4 N N 1 1 1 1 1 1 b b 2 2 Y Y ( ( ) ) b b 2 2 = = i i /2 /2 Y Y , , ( ) Y ( ) Y Y Y Y Y e e H H e e H H Y Y 2 2 b b i i i i y y y reli y reli . . c m n . . c m n n n y y y reli y reli . . c m n . . c m n n n y y 2 2 ! ! . . c m n . . c m n y = = 2 2 ! ! y rel rel n n 2 2 i i i i
Walter Kohn (1923-2016) Generalized Kohn theorem: The frequency of resonant absorption of long- wave radiation from a pair-interacting electron gas localized in a parabolic QD doesn't depend on the number of particles. In other words, single-particle transitions are realized in a multiparticle system.
Hamiltonian for N noninteracting electrons in an asymmetric QD, its diagonalization and energy spectrum 2 2 j j p p 1 1 1 2 2 1 2 2 1 N N N N 1 2 2 ( ) ( ) + + + + = = + + + + + + = = + + + + 2 2 2 2 H H C C C C C C C C 2 x x 2 j j 2 y y 2 j j H H m m x x y y 0 0 x x x x x x y y y y y y 0 0 2 2 m m = = = = 1 1 1 1 j j j j N N 1/2 1/2 = = m m p p ( ) ( ) + + + + + + C C c c + + = = + + ( ) x y ( ) x y , ( ) j x y , ( ) j x y x y = = c c c c ( ( ) ) c c x y x y im im ( ) x y ( ) x y , ( ) j x y , ( ) j x y , ( ) j x y , ( ) j x y j j j j , ( ) j x y , ( ) j x y , ( ) j x y , ( ) j x y 2 2 = = ( ) x y ( ) 1 1 j j 1 1 1 2 2 1 2 2 = = + + + + + + E E n n n n , , n n n n x x x x y y y y x x y y
Hamiltonian of the interacting system, electron-electron interaction and commutation relations 2 2 e e N N = = = = + + ( ) u r ( ) u r H H H H U U = = (1,..., (1,..., ) ) ( ( ) ) U U N N u r u r r r i i j j 0 0 r r i j = i j = 1 1 + + + + + + + + = = 0 0 = = 0 0 0 0 0 n n x n n 0 n n x n n 0 n n x n n 0 n n x n n , , (1) (1) H C H C H C H C C H C H C C 0 0 , , 0 0 , , , , , , x x x x x x a a x x y y y y y y y y x x x x 0 0 0 n n x n n 0 n n x n n E E + + + + 0 0 0 n n x n n 0 n n x n n (1) (1) C C E E , , , , , , , , , , x x a b a b y y y y x x y y y y x x = = U = U N C N C = , , (1) (1) H C H C C C (1,..., (1,..., ), ), 0 0 , , , , , , , , x y x y a b a b x y x y x y x y
System in an electric field, commutation relation H H = = + + H H H H 0 0 i t i t = = (cos ,sin ) (cos ,sin ) = = ( ) ( ) E t E t e e E E ( ) ( ) H H e e r r E t E t j j 0 0 j j 1/2 ) ( ) ) ( ) ( ( 1/2 + + = = + + / / x x m m C C C C j j x x x x x x j j H C H C H C H C = = + + = = + + , , , , , , , , H C H C H H H C H C , , 0 0 , , 0 0 , , , , x y x y x y x y x y x y x y x y 1/2 ) ( ) ( ) ( ) ( 1/2 + + = = + + / / y y m m C C C C j j y y y y y y j j H C H C = = = = = = , , 0 0 , , , , (1) (1) H C H C H C H C C C , , , , 0 0 , , , , , , x y x y x y x y x y x y a b a b x y x y
Obtained results . Electron gas total energy dependency from QD semiaxis b for different interaction parameter 1(2) 1(2)
Obtained results 1(2) Electron gas total energy dependency from QD semiaxis c for different interaction parameter 1(2)
Obtained results Electron gas center of mass energy diagram
Obtained results Electron gas relative energy diagram
Obtained results Electron gas total energy diagram
Obtained results Electron gas total energy diagram
Main Results: 1. In the case of a strongly oblate asymmetric ellipsoidal quantum dot, the confining potential of the slow subsystem is described within the framework of a two-dimensional asymmetric parabolic well. 2. In the quantum dot under consideration, for the case of a pair- interacting electron gas, the conditions for fulfilling the generalized Kohn theorem are realized. 3. The resonance frequencies of the transitions described by the generalized Kohn theorem depend on the geometric parameters of the ellipsoidal quantum dot and can be controlled by changing these parameters.
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