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In this work we used the wrapper method to solve the problem of interaction of electrons and hybridons in a free quantum wire. Using the Dirac turbulence theory, we established the expression for determining the rate of scattering and the recovery time for the electrons in the wire.
Scientific Journal − No27/2018 107 INTERACTION OF ELECTRONS AND HYBRIDONS IN A FREE QUANTUM WIRE Ta Anh Tan1, Đang Tran Chien2 Pham Van Quang3 Hanoi Metropolitan University University of Natural Resources and Environment Vietnam Commander officer training college Abstract: In this work we used the wrapper method to solve the problem of interaction of electrons and hybridons in a free quantum wire Using the Dirac turbulence theory, we established the expression for determining the rate of scattering and the recovery time for the electrons in the wire Keywords: Hybridons, rate of scattering, recovery time, turbulence theory Email: tatan@hnmu.edu.vn Received 22 September 2018 Accepted for publication 15 December 2018 INTRODUCTION In the publication [1], we have linear combinations of the oscillations in the quantum wires that are pair of the LO oscillators, IP1 and IP2 All of these oscillation modes vibrate at the same frequency and vector Quantization leads to the concept of a new quantum that is hybridon hybrid The interaction of electrons with these hybrid particles is described as internal and external scattering in an infinite quantum well Using the cone method we solve the problem for electrons in quantum wires Then using the Dirac turbulence theory we established the expression for determining the rate of scattering and the recovery time for the electrons in the wire CALCULATIONS 2.1 The state of the Electron in the quantum wire Electrons moving in quantum wires are influenced by crystal circuitry and captive power The wave energy and energy of the electron in the quantum wire are the solution of the Shrödinger equation Ha Noi Metroplolitan University 108 −ℏ 2 m ∇ + U (r ) + V (r ) Φ (r ) = E Φ (r ) The equation of the above equation is found by the effective mass-mass method [2] Expressions for electron wrappers in quantum wires: F ( r, ϕ , z ) = eik z z eilϕ J (k r ) (π R02 L ) J m+1 (κ mn ) m mn (1) and the energy of the electron in the quantum wire Erϕ = Emn = with k m ,n = ℏ2 ( k 2z + k mn2 ) m* κ mn r 2.2 Probability of state transition When studying the interaction of electrons with phonons, as well as the interaction of electrons with other particle norms in solids, we need to study the probability of electron state transfer under the effect of the small V(t) M ml = 2π m V (t ) l ℏ with: M i , f = a f ( t ) = aif ( t ) 2 δ ( Em − El ) (2) (i - is the initial state symbol) [5] Such turbulence is responsible for the transfer of the system from one quantum state to another Electrons in solids are granular and occupy single-electron states in the energy-domain structure They are described by the Block function, which is the area index, k is the wave vector, the spin of the electron In this section we only care about the electrons in the conduction band, so the region index only appears in some cases Furthermore, when the transfer in the spinconducting region of the electron is generally preserved, then we write the state function of the electron normally through its wave vector Phonon is a particle standard that describes network oscillations The number of phonons of the individual states is characterized by the observable wave vectors and the j-branches of the diffusion spectrum ω j ( q ) The electron-phonon interaction is expressed by the phonon generation or phonon removal (q, j) with the simultaneous transformation of the electron state k , σ to the state k ± q, σ We now determine the probability of electron transfer by the optical oscillator Scientific Journal − No27/2018 109 The probability of state displacement is determined by the formula (2), where the disturbance is replaced by the Hamiltonian interaction between the electron and the phonon optical The initial states i and f are characterized by the number N(q) of the phonon and the k-wave vector of the electron i = k i , N ( q ) ; f = k f , N ('q ) The state after absorbing an optical phonon (the end state of the process) is given by f = k f , N ( q ) − and I have k f = k i − q , E f = Ei + ℏω ( q ) The probability of state transition for phonon absorption is given by [3] M i , f = M k i + q ,k i = M + ( k , q ) = 2π k i + q, N ( q ) H int k i , N ( q ) − δ ( E f − Ei − ℏω ( q ) ) (3) ℏ Status after the emission of an optical phonon (end state of the process) given by f = k f , N(q) + and I have k f = k i − q , E f = Ei − ℏω ( q ) The probability of state transition for phonon emission is given by: The probability of state transition for phonon emission is given by: M i , f = M k i − q ,k i = M + ( k , q ) = 2π k i + q, N ( q ) + H int k i , N (q ) ℏ δ ( E f − Ei + ℏω ( q ) ) (4) 2.3 Rate of scattering in quantum wires From the theory for mass semiconductor we apply to calculate the scattering speed for quantum wires Here the wire system is a one-dimensional system so that the state of the electron and the phonon optical are only represented by the wave vector in the z axis of the wire From (3), (4), the probability that the electron's energy level in wire from i-state to end-state in a time unit is determined as follows: M i→ f = 2π k zf , N ('q ) H int N (q ) , k iz ℏ δ ( ETf − ETi ± ℏωsp ) (5) where M i → f is the scattering rate of the electron from the i-state to the f-state, N (q ) and N('q) are the phonon distributions in the absorption and phonon emission, according to the Bose-Ensten distribution, H int is the Hamiltonian interaction of electrons and phonons ETf , ETi is the energy of the electron at state x and y with: ETi = ℏ2 ℏ2 i f ; k + k E = k m2 , n + ( k zf ( ) m , n z T * * 2m 2m ( ) ( ) ) (6) Ha Noi Metroplolitan University 110 2.3.1 Hamiltonian interaction in the wires The electron-phonon interaction in the wire is Fröhlich's interaction, so the Hamiltonian interaction in the wire is defined as: H = -eΦ + A P (7) The scalar Φ is related to mode LO and the vector A is related to mode IP and P = −i e ℏ∇ With me is the weight of the electron, the scalar Φ is related to mode LO and me the vector A is related to mode IP In this case, the scalar Φ is connected to the LO mode and the vector A is connected to the IP mode and P is the operator Where me is the mass of the electron, the scalar Φ is related to the LO oscillation mode and the vector A is associated with the oscillation mode IP 2.3.2 The scalar Φ The scalar Φ is defined in the relation only through the LO mode in the following way: EL = ρ 0u L = -gradΦ Inside ρ = (8) 1 1 e* ; e* = MV0ω L2ε 02 − in the cylindrical coordinates of the ε 0V0 ε∞ ε0 expression gradΦ is given by the expression: gradΦ = e r ∂Φ ∂Φ ∂Φ + eϕ + ez ∂r r ∂ϕ ∂z (9) We obtain the following equations: ∂Φ ∂r = Aρ0 iq Ls, p qz eisϕ eiq z z J 's (q sL, p r ) ∂Φ s isϕ iq z z = − Aρ0 e e J s (q sL, p r ) qz r r ∂ϕ ∂Φ ∂z = − Aρ0 eisϕ eiq z z J s (q sL, p r ) From these equations, identify the Φ : (10) Scientific Journal − No27/2018 Φ = Aρ iq sL, p qz Φ = − Aρ 111 eisϕ eiq z z ∫ J 's (q sL, p r )dr s iq z z e J m (q sL, p r ) ∫ eisϕ dϕ qz (11) Φ = − Aρ0 eisϕ J s (q Ls , p r ) ∫ eiq z z dz or: Φ = Aρ0 i isϕ iq z z e e J m (q sL, p r ) qz (12) Substitution of the normalization coefficient, we get the expression of scalar potential 2 M ρ0ω η − s I s ( q z R0 ) isϕ iq z z Φ= Xe e J s (q Ls , p r ) π LΘ q 2z R0 η J s ( q sL, p R0 ) (13) 2.3.3 Potential vector In the wire, the vector is determined by the IP mode according to the following formula: − ∂A = E = ρp u p ∂t (14) whit: q 2Z R 02 + s η − s I 2s ( q z R ) u rp = ρ p B I s ( q z r ) e isϕ e i q z z 2 q Z R0 η 2 p η − s I s ( q z R ) I ( q r ) e isϕ e iq z z u ϕ = is ρ p B s z q 2Z R 02η I s ( q z R ) 2 u p = i ρ B η − s I m ( q z R ) I ( q r ) e isϕ e iq z z z p s z q z R 0η I s ( q z R ) in it, ωsp2 − ρ p = ρ0 Identify the integral: ε∞ ω ε0 L ε∞ − ωL ε0 (15) Ha Noi Metroplolitan University 112 q 2Z R02 + s η − s I s2 ( q z R0 ) Ar = i ρp B I s (q z r )eisϕ eiq z z 2 ωq Z R0η η2 − s 2I 2s ( q z R0 ) ρ A = − s B I s (q z r )eisϕ eiq z z ϕ p 2 ω η q R I q R Z s ( z 0) 2 A = − ρ B η − s I s ( q z R0 ) I (q r )eisϕ eiq z z p z ωq z R0η I s ( q z R0 ) s z (16) Substituting the standardized coefficients into ones: q 2Z R02 + s η − s 2I 2s ( q z R0 ) M Ar = i ρ p ωX I s (k z r )eisϕ eiq z z 2 π ω L Θ R q η Z 2 η − s I s ( q z R0 ) M ωX 2 I s (q z r )eisϕ eiq z z Aϕ = − s ρp π LΘ ωq Z R0η I s ( q z R0 ) η − s 2I 2s ( q z R0 ) M A = −ρ ω X I s (q z r )eisϕ eiq z z p z π ω η Θ L q R I q R z s ( z 0) (17) 2.3.4 Hamiltonian interaction Momentum P is defined as follows: P=- iℏe iℏe ∂ ∂ ∂ ∇=+ eϕ + ez er me me ∂r ∂ϕ ∂z (18) ie ∂ ∂ ∂ + Az Φ Ar + Aϕ ∂r ∂ϕ ∂z me (19) Have: H = -eΦ - Find the Hamiltonian interaction as follows: H = −e∑ Ξ s, p M X e isϕ e iq z z π L q z R 2η ρ ω R 0η L J q L R J s (q s , p r ) + s ( s, p ) ∂ 2 − i q Z R + s I s ( q z r ) ∂ r + ρ i ℏ p + m q z R 0η sη ∂ ∂ e + I (q r ) + I (q r ) I (q R ) s z ∂ϕ I (q R ) m z ∂z s z s z (20) put: H int = −e∑ s, p ℏ ⌢ ⌢ ℚ {a + + a } 2π Lω (21) Scientific Journal − No27/2018 113 Inside: iℏ ρ p e isϕ e iq z z ρ 0ω R0η ℚ=Ξ J s (q sL, p r ) + L q z R0 η J s ( q s , p R ) me ∂ 2 − i q Z R0 + s I s (q z r ) ∂ r + q z R0η sη ∂ ∂ + I ( q R ) I s (q z r ) ∂ϕ + I ( q R ) I m (q z r ) ∂z s z s z (22) 2.3.5 Scattering speed From (2) and (5) we have: Mi→ f 2π ℏ ⌢ ⌢ mnk zf , N( q)' − e ℚ{a + + a} N(q) , mnkiz = ∑ ℏ 2π Lω s, p δ ( ETf − ETi ± ℏωsp ) (23) An intrasubbling scattering implies that one electron from the beginning state absorbs There are no electrons N ( q ) and or emits one phonon and moves to the final state mnk zf N '( q ) is the function of the phonon distribution in phonon delivery and absorption Here we consider multiple systems so they follow the Bose-Einstein distribution The quantum transfer probability in (5) will be determined: For phonon absorption we have: Mi, f = e 2 N(q) mnk Lωsp f z ∑ℚ mnk δ ( ETf − ETi − ℏωsp ) i z s, p (24) For the phonon emission process we have: Mi, f = e N(q) + Lωsp mnk f z ∑ℚ mnk s, p i z δ ( ETf − ETi + ℏωsp ) (25) matrix element G i , f = mnk zf ∑ ℚe ϕ e is iq z z mnk zi s, p Inside: m nk f z 2 − im ϕ − i k zf z = e J m (kr )e π L R J m + (κ m n ) (26) Ha Noi Metroplolitan University 114 mnk i z 2 imϕ ik iz z = J n (k mn r )e e 2 π LR0 J m +1 (κ mn ) (27) Instead (27) to (26) we have: Gi , f = ∑ 2 s,p π LR0 J m+1 (κmn ) R L 2π ∫∫ ∫ J f m i (k mn r )e−imϕ e−ik z z ℚeisϕ eiqz z J m (k mn r)eimϕ eik z z rdrdϕdz (28) 0 put: R £1 = ∫ rJ s (q r )J (k mn r )dr; L s, p m R £ = ∫ I s (q z r )J 2m (k mn r )dr R R 0 (29) £3 = ∫ rJ m (k mn r )I s (q z r )J m+1 (k mn r )dr; £ = ∫ rI s (q z r )J 2m (k mn r )dr it will be obtained: G i, f = ∑ p ρ ω R η ℏρ 2Ξ 0 £1 + p 42 L q z ηR0 m+1 (κ mn ) J ( q p R0 ) me 2 k iz q z R0 η m q R £ − k q R £ − £4 Z mn Z I ( q z R0 ) (30) We consider, in approximate terms, the contribution of the first solution of the Bessel function to the largest for the Hamiltonian interaction, and to examine the intrasubband scattering for the electrons in the lowest energy region ie the regional index m = and n = 1: G i, f = ρ ω R η ℏρ p k iz q z R0η 2Ξ 2 0 £ − k q R £ − £4 01 Z q zη R02 J12 (κ 01 ) J ( q0L p R0 ) me I ( q z R0 ) (31) Inside: L L 2 M ω01 ( q 01 ) J1 (q 01 R ) + ε ( ω ) ρ q Z R + q z Rη I (q R ) Ξ= + p z L J 20 ( q 01 I 02 ( q z R ) η2 R) V0 q z η = I ( k z R0 ) − R0q z I1 ( q z R0 ) − (32) (30) into (24) and (25) we will find the electron scattering rate determining method for the phonon absorption and emission states as follows: For phonon absorption we have: Scientific Journal − No27/2018 Mi+, f = e2 N( q) Lω01 115 Gi , f δ ( ETf − ETi − ℏω01 ) (33) For the phonon emission process we have: N(q) +1 Gi, f δ ( ETf − ETi + ℏω01 ) (34) δ ( ETf − ETi ∓ ℏω01 ) = δ ( ETf − ETi ∓ℏω01 ) (35) Mi−, f = e2 Lω01 With delta function: We find the general expression for the recovery time: τ if = e2 2πω012 ∫ Gi, f ℏ2 i i 2 − N δ (q) * ( k z + q z ) − ( k z ) + ℏω0 p + 2 2m dq z + N + δ ℏ k i + q − k i − ℏω ( z ) p * ( z z) (q ) 2m ( ) To integrate by qz we proceed as follows: From 2 ℏ2 ℏ2 i k + q = k i ± ℏω01 ( ) z z * * ( z) 2m 2m Or (36) ℏ 2 2ℏ i qz ± k z q z cosϑ ∓ ℏω01 = ϑ is the angle between qzand kzi m* m* According to [4] we have: q = −k cos ± k 2cos 2ϑ + x z z z (1) q z ( 2) = k z cos ± k 2z cos 2ϑ − x (37) ℏ x2 = ℏω01 m* (38) Put: Pay attention to the distribution function of the phonon: N (q ) = N( q ) k BT ℏω 01 ℏω ≈ N(q) + = − 01 kT ≈ e k BT ℏωB 01 −1 e k BT >> ℏω01 (39) k BT > ℏω01 ) or is k z >> x , We have: q z ( ) = 0; q z ( max ) = 2k iz (40) Because of: kBT >> ℏω01 N( q ) = N ( q ) ≈ N ( q ) + = m* = τ if 2πω012 8π ℏ 2ω01k iz e2 k iz ∫ ( N( ) + 1) G q k BT >> ℏω01 q3z dq z attention to i, f m*e2 kT = 3 i3 B τ if 8π ℏ ω01k z ℏω01 k iz ∫ 2k BT kT +1 ≈ B ℏω01 ℏω01 G i , f q3z dq z (41) At low temperature ( k BT