Interaction energy between electrons and longitudinal optical phonon in polarized semiconductor quantum wires

In this work we calculated the displacement of the lattice nodes in the quantum wire. Then building Hamilton''s interaction between electrons and phonons in quantum wires and calculating dispersion expressions. Draw dispersion curves for modes p = 0 and wire with radius of 100 Å and 150 Å.

TRƯỜNG ĐẠI HỌC THỦ ĐÔ H 74 NỘI INTERACTION ENERGY BETWEEN ELECTRONS AND LONGITUDINAL OPTICAL PHONON IN POLARIZED SEMICONDUCTOR QUANTUM WIRES Dang Tran Chien Hanoi University of Natural Resources & Environment Abtract: Abtract: In this work we calculated the displacement of the lattice nodes in the quantum wire Then building Hamilton's interaction between electrons and phonons in quantum wires and calculating dispersion expressions Draw dispersion curves for modes p = and wire with radius of 100 Å and 150 Å Built an energy expression that interacts between the electron and the longitudinal optical phonon in polarized semiconductor quantum wires Keywords: Longitudinal optical phonon, dispersion expression, quantum wires, Hamilton Email: dtchien@hunre.edu.vn Received 27 April 2019 Accepted for publication 25 May 2019 INTRODUCTION Nowadays, with modern techniques in crystal culture, many material systems have been created with nanostructures Low-dimensional system structure not only significantly changes many properties of materials, but also appears many new physical properties superior compared to conventional three-dimensional electron systems Electrons and their vibrations are distorted because they become low-dimensional and low symmetry One of the methods for making quantum wires is to create alternating thin semiconductor layers These semiconductor classes have different band gaps Then by etching such as chemical corrosion, corrosion of plasma, we have a quantum wire There have been many authors studying on quantum wires in the world Longitudinal optical oscillations (LO) and electron transfer rate are calculated in [5, 6], the electron scattering rate in rectangular quantum wire in [1-4] etc… In semi-conductive material polarized conductors (Polar-semiconductor), electrons mainly interact with longitudinal optical phonons But the energy interaction between electrons and phonons in quantum wires has not been studied much TẠP CHÍ KHOA HỌC − SỐ 31/2019 75 So this paper will focus on calculation of energy interaction between electrons and longitudinal optical phonons in GaAs/AlGaAs polarized semiconductor quantum wires CALCULATIONS 2.1 Oscillations in a quantum wire Here the cylindrical coordinate system was applied [∇ + ki2 ]u L = (1) Where uL was denoted as longituadinal oscillation  ∂ ∂2  ∂2 ∂2 + + + + k i2  u L =  2 2 r ∂ϕ ∂z  r ∂r ∂r  (2) The solution of the equation (2) was written as follows: u ( L ) (r , ϕ , z ) = A.u ( L ) (r ).eipϕ eiqz z e − iωt (3) Put (3) into (2) we have: ∂2 ( L) ∂ ( L) p2 ( L) u ( r ) + u ( r ) − u (r ) − q z u ( L ) (r ) + k i u ( L ) (r ) = 2 ∂r r ∂r r (4) Due to differential equation (4) has only r variable so we have:  p2 d ( L) d ( L) 2  (L) u ( r ) + u ( r ) +  − − q z + k i  u (r ) = dr r dr  r  q i2 = k i2 − q z = (ω L2 − ω ) β −2 − q z  p2  ( L) d (L) d (L) u ( r ) + u ( r ) +  qi −  u (r ) = dr r dr r   (5) Put χ ps = q i r , equantion (5) changes into d ( L) d p ( L) (L) u (r ) + u (r ) + (1 − )u (r ) = d χ 2ps χ ps d χ ps χ ps (6) J p ( χ ps ) in the solution to the modified Bessel’s equation is referred to as a modified Bessel function of the first kind The second material area, the solution of the second region is Hankel function H p ( χ ps ) = J p ( χ ps ) + iN p ( χ ps ) TRƯỜNG ĐẠI HỌC THỦ ĐÔ H 76 NỘI For the first material area, the solution was: u ( L ) (r , ϕ , z ) = AJ p ( χ ps ) eipϕ eiqz z e −iωt Longituadinal Optical (LO) mode satisfies the condition: [∇.u] = In cylindrical coordinate system [∇.u] =   ∂ (L) ∂ ∂ ∂  u z (r ,ϕ , z ) − ( ruϕ ( L ) (r ,ϕ , z ) )  er +  u r ( L ) (r , ϕ , z ) − u z ( L) (r , ϕ , z )  eϕ +  r  ∂ϕ ∂z ∂r  ∂z    1 ∂ ∂ ( L) u r (r , ϕ , z )  e z = +  ( ruϕ ( L ) (r , ϕ , z ) ) − r  ∂r ∂ϕ  (7) The axial unit vectors are linearly independent for each other so from (7), we obtain the following system of equations: ∂ ( L)  ∂ ( L)  ∂ϕ u z ( r , ϕ , z ) − r ∂z uϕ (r , ϕ , z ) =  ∂ (L)  ∂ (L)  u r (r , ϕ , z ) − u z (r , ϕ , z ) = ∂r  ∂z ∂ ( L) ∂ ( L)  (L) uϕ (r , ϕ , z ) + r ∂r uϕ (r , ϕ , z ) − ∂ϕ u r (r , ϕ , z ) =  (8) With some calcultion we obtained the equation for the ion displacement of the LO mode in the quantum wire as follows:  1L −iq1 ipϕ iq z z − iωt ' u r = q A z J p ( q1r ) e e e z  p  1L A z J p ( q1r ) eipϕ eiqz z e−iωt (9)  uϕ = rq z  u1L = A J ( q r ) eipϕ eiqz z e−iωt z p  z  For the second material area, the motion equation for the node is also satisfied as for region At the same time, the Hankel function with the derivative is completely similar to the Bessel function, so we have:  L −iq2 ' ipϕ iq z z − iω t u r = q A z H p ( q2 r ) e e e z   2L p A z H p ( q2 r ) eipϕ eiqz z e−iωt uϕ = rq z  L u = A H ( q r ) eipϕ eiqz z e−iωt 2z p  z  (10) TẠP CHÍ KHOA HỌC − SỐ 31/2019 77 2.2 The dispersion equations Applying continuous boundary conditions that is the perpendicular velocity component ie the direction of r continuously and the pressure at the interface is continuous Let β = β1 where β1 , β sound velocity parameters in the material area 1, ρ1 , ρ2 β2 denote mass density in the area of material 1,2 If we let: A1z = A(1) , A2 z = A(2) ; ρ = ρ1 ρ2 Applied the continuous conditions of pressure at the interface, we have β ρ ∇ u1L r = R = ∇ u2 L r = R (11) Put equations (9), (10) into (11), we obtained: ∂ ∂ 1L ∂ ∇.u1L = u1rL (r , ϕ , z ) + u1rL (r , ϕ , z ) + uϕ (r , ϕ , z ) + u1zL (r , ϕ , z ) r ∂r r ∂ϕ ∂z ∇.u1L r = R0  p2  q  q  ipϕ iqz z − iωt ( ) = −iA(1)  J p'' (q1R0 ) + J ' p (q1 R0 ) −  + q J q R e e e  z p q z R0  q z R0   q z  Do the same for the second area we have:  ''  p2  q2  (2)  q2 ' ( ) ( ) = − iA H q R + H q R − + q z  H p (q2 R0 )  eipϕ eiqz z e− iωt   r = R0 p p 2 q z R0  q z R0   q z  from (11) one gets ∇.u L  q12 ''   p2  q + q z  J p (q1 R0 )  = J p (q1 R0 ) + J ' p (q1 R0 ) −  q z R0  q z   q z R0  β ρ A(1)   q2 ''   p2  q2 ' =A  H p (q2 R0 ) + H p (q2 R0 ) −  + q H ( q R )  z  p q z R0  q z R0   qz  (2)  q12 ''   p2  q J p (q1 R0 ) + J ' p (q1 R0 ) −  + q z  J p (q1 R0 )  = q z R0  q z   q z R0  β ρ A(1)   q   p2  q = A(2)  H ''p (q2 R0 ) + H ' p (q2 R0 ) −  + q z  H p (q2 R0 )  q z R0  qz   q z R0  (12) From the continuous velocity conditions, we put the expressions (9) and (10) into ρ1−1/ 2uɺ 1rL r = R0 = ρ 2−1/ 2uɺ 2r L r = R0 (13) TRƯỜNG ĐẠI HỌC THỦ ĐÔ H 78 NỘI Then taking the simple derivative and transformation, we have: ρ A(1)q1 J 'p (q1 R0 ) − A(2)q H ' p (q R0 ) = (14) from (12) and (14), we have a equation system as below:   ''   p2  q1 ' (1) q1 J p (q1 R0 ) −  J q R + q ( )  β ρ A  J p (q1 R0 ) +  −  z p qz R0  qz R0    qz     p2  q2  (2)  q2 '' ' H p (q2 R0 ) + H p (q2 R0 ) −  + q H ( q R ) =0 − A   z p   qz R0  qz R0   qz     A(1)q1 J p' (q1 R0 ) − A(2)q2 H ' p (q2 R0 ) =  ρ  (15) From equation (14), we have: A(2) = τ= A(1)q1 J p' (q1 R0 ) q H ' p (q R0 ) ρ q1 J p' (q1 R0 ) q H ' p (q R0 ) ρ It leads to A(2) = A(1)τ For the equation (15) to have a non-trivial solution, its determinant must be zero ie: β2   q12 ''   q2 '' q H p (q2 R0 ) + H ' p (q2 R0 ) −   J p (q1 R0 ) +   qz qz R0   −  qz ρ      p   p  q  =0  + J ' p (q1 R0 ) −   − + q ( q ) J R q + z  p z  H p (q2 R0 ) 2  qR   qR  q R z  z       z  q1 J p' (q1 R0 ) −q2 H ' p (q2 R0 ) ρ Use the properties of the Bessel function and transform, we have the dispersion expression for the (LO) mode as follows: ρ q2 (ω12 − ω ) J p (q1 R0 ) H ' p (q2 R0 ) = q1 (ω22 − ω ) H p (q2 R0 ) J p' (q1 R0 ) (16) Applied (16) for a quantum wire GaAs/Al0.3Ga0.7 As with parameters as followings: ω1 = 292.8cm−1 ; ω2 = 0.95ω1 ; β1 = 4.73 ×103 ms −1 ; β = 1.06β1 ; ρ = 1.11 TẠP CHÍ KHOA HỌC − SỐ 31/2019 79 For mode p=0, the quantum wire with radius Ro=100 Å, 150 Å, the dispersion curves were shown in Fig.1a and Fig 1b, respectively a) b) 0.98 0.98 0.97 0.96 0.96 0.95 w/w1 w/w1 0.94 0.92 0.94 0.93 0.92 0.90 0.91 0.90 0.88 q z R , F or mode p=0, R =100A o 10 11 10 11 qzR0 , for mode p=0, R o =150 Ao Fig The dispersion curves of the quantum wire with radius Ro 150 Å (a) and 100 Å (b) As can be seen from Figure 1, in the wire with radius Ro=100 Å, the phonon energy is quantized and separated into energy levels farther apart In the wire with radius Ro=150 Å, the energy is separated into close levels 2.3 Potential interaction LO oscillation generated electric field which was calculated by the formula: E = − gradφ (17) In the our situaion, electric field was written as below: E = − ρ oi u iL Where ρoi (18) denote bulk charge density in material area i, uiL denote equations in (9), (10) In cylindrical coordinate, gradϕ can be written as: gradφ = er ∂φi ∂φ ∂φi + eϕ + ez i ∂r r ∂ϕ ∂z (19) From (17), (18) and take note of (19), we have: ∂φi ∂φ ∂φi + eϕ + e z i = ρ 0i u iL ∂r r ∂ϕ ∂z i = 1, er Put equation (9) into (20), we obtained equations for area (20) TRƯỜNG ĐẠI HỌC THỦ ĐÔ H 80 ∂φ1 ∂r = A(1) ρ 01 iq1 ipϕ iqz z − iωt ' e e e J p (q1 r ) qz ∂φ1 p ipϕ iqz z − iωt = − A (1) ρ 01 e e e J p (q1 r ) r ∂ϕ qz r ∂φ1 ∂z NỘI (21) = − A (1) ρ 01eipϕ eiqz z e − iωt J p (q1 r ) Taking integrals and transformations we have the potential interaction for the material area as follows: φ1 = A(1) ρ 01 i J p (q1 r )eipϕ eiq z z e − iωt (22) qz Completely similar, we can calculate the interaction potential for material area as follows φ2 = τρ02 A(1) i H p (q r )eipϕ eiqz z e− iωt qz (23) 2.4 Hamilton interaction We determine Hamilton's interaction between electrons and phonon in the form of Fröhlich: (24) H int = −eφ Put equation (22) and (23) into equation (24), we have a Hamiltonian equation as follows: Η int i ⌢  iPϕ iq z z − iω t ⌢ + −eAρ 01 q J p (q1 r )e e e {a + a}  z = −eAτρ i H (q r )eipϕ eiqz z e− iωt {a⌢ + + a⌢} 02 p  qz Η int = −eA i qz rR0  ρ01 J p (q1r )θ ( R0 − r ) +  iPϕ iq z − iωt ⌢ + ⌢   e e z e {a + a} + H ( q r ) r − R τρ θ (  02 p )  (26) 0 when r>R0 ; 1 when r (30) E (0) − E k(0),1> 0,0 Xét k , > is the status where electrons in status with m, n, kz and phonon in the status p,s,qz that contributed into E0(2) So the equation (30) becomes to (2) E =∑ < 0, Η int k ,1 > (31) E (0) − E k(0),1> 0,0 Where k , ≡ k Put (27) into (31), one can get: (2) E =∑ ⌢ ⌢ < 0, ℤ{a + + a} k ,1 > E (0) − E k(0),1> 0,0 =∑ ⌢ < 0, ℤa + k ,1 > E (0) − E k(0),1> 0,0 +∑ ⌢ < 0, ℤa k ,1 > E (0) − E k(0),1> 0,0 It is easy to see that the first term of (32) is zero The second term becomes to (32) TRƯỜNG ĐẠI HỌC THỦ ĐÔ H 82 ∑ ⌢ < 0, ℤa k ,1 > E (0) − E k(0),1> 0,0 < 0, ℤk , > =∑ NỘI E (0) − E k(0),1> 0,0 So (32) can be written as follows: E =∑ (2) < 0, ℤk , > (33) E (0) − E k(0),1> 0,0 T = < 0, ℤ k , > (34) R0 χ 01 χ (1) χ − ms ρ J ( r ) J ( r ) J m ( mn r )rdr + m 01 ∫ R0 R0 R0   2eA T =  ∞ χ 01 χ (2) χ  k z R0 J1 (χ 01 ) J m +1 (χ mn )  +τρ − ms ( ) ( J r H r ) J m ( mn r )rdr m 02 ∫ R0 R0 R0 R0 Where (q ) (q ) mn = (ω12 - ω ) β1−2 - k 2z 2 mn = (ω22 - ω ) β 2−2 - k 2z Electron energy in a quantum wire can be expressed as below: ET = ℏ 2k 2z ℏ 2χ mn + 2m* 2m*R0 (34) Electron energy in the fundamental status is E (0) = 0,0 ℏ χ 01 2m * R0 (35) Electron-phonon energy at status k ,1 E (0) = k ,1 ℏ2χ ℏ 2k 2z + mn + ℏω 2m * R0 2m * (36) With { 1/ } ω = ω12 - β12 ( q12 ) mn + k 2z  If we let denominator of equation (33) be MS (37) TẠP CHÍ KHOA HỌC − SỐ 31/2019 MS z =ℏ k + 2m * 83 ℏ2 χ − χ 01 + ℏ ω12 - β12 ( q12 )mn + k 2z  ) ( mn 2m * R0 { 1/2 } (38) We can get the energy regulation as follws: R0 χ 01 χ (1) χ − ms ρ J ( r ) J ( r ) J m ( mn r )rdr + 01 ∫ m R0 R0 R0   2eA   ∞ χ χ (2) χ  k z R0 J1 (χ 01 ) J m +1 (χ mn )  +τρ 02 ∫ J ( 01 r ) H m ( − ms r ) J m ( mn r )rdr R0 R0 R0 R0 E0(2) = ∑ m, n , s ,k z ℏ 2k 2z ℏ2 χ − χ 01 + + ℏ ω12 - β12 ( q12 )mn + k 2z  ) ( mn 2m * 2m * R0 { 1/ } Finally we have the interactive energy of the electron and phonon in the quantum wire ℏ χ 01 T E= + ∑ 2 2 2m * R0 m ,n , s ,k z ℏ k z + ℏ ( χ 2mn − χ012 ) + ℏ ω12 - β12 ( q12 )mn + k 2z  2m * 2m * R0 { 1/ } Thus the energy of the electrons in the quantum wire is quantized and depends on the kz vector of the electrons along the Oz axis and the radius of the wire CONCLUSIONS We have sussesfully calculated the displacement of the lattice nodes in the quantum wire Thereby building Hamilton's interaction between electrons and phonons in quantum wires and calculating dispersion expressions Drew dispersion curves for modes p = and wire with radius of 100 Å and 150 Å Constructing an energy expression that interacts between the electron and the longitudinal optical phonon in polarized semiconductor quantum wires ACKNOWLEDGEMENTS This research is a result from the HURE project 2019 without expense of the government budget REFERENCES Babiker M., Ridley B K (1986), "Effective-mass eigenfunctions in superlattices and their role in well-capture", Superlattices and Microstructures 2, pp.287-291 Constantinou N., K Ridley B (1990), Interaction of electrons with the confined LO phonons of a free-standing GaAs quantum wire 84 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI Constantinou N C (1993), "Interface optical phonons near perfectly conducting boundaries and their coupling to electrons", Physical Review B 48, pp.11931-11935 Ridley B K (1994), "Optical-phonon tunneling", Physical Review B 49, pp.17253-17258 W Kim K., Stroscio M., Bhatt A., Mickevicius R V., Mitin V (1991), Electron@optical@ phonon scattering rates in a rectangular semiconductor quantum wire Wiesner M., Trzaskowska A., Mroz B., Charpentier S., Wang S., Song Y., Lombardi F., Lucignano P., Benedek G., Campi D., Bernasconi M., Guinea F., Tagliacozzo A (2017), "The electron-phonon interaction at deep Bi2Te3-semiconductor interfaces from Brillouin light scattering", Scientific Reports 7, p16449 NĂNG LƯỢNG TƯƠNG TÁC GIỮA ELECTRONS VÀ PHONON QUANG DỌC TRONG DÂY LƯỢNG TỬ BÁN DẪN PHÂN CỰC Tóm tắ tắt: Trong báo chúng tơi tính độ dịch chuyển nút mạng dây lượng tử, từ xây dựng Haminton tương tác điện tử phô nôn dây lượng tử tính biểu thức tán sắc Vẽ đường cong tán sắc cho mode p = dây với bán kính 100 Å 150 Å Xây dựng biểu thức tính lượng tương tác điện tử LO dây lượng tử bán dẫn phân cực Keywords: Phonon quang dọc, biểu thức tán sắc, dây lượng tử, hàm Hamilton  ... calculation of energy interaction between electrons and longitudinal optical phonons in GaAs/AlGaAs polarized semiconductor quantum wires CALCULATIONS 2.1 Oscillations in a quantum wire Here the cylindrical... nodes in the quantum wire Thereby building Hamilton's interaction between electrons and phonons in quantum wires and calculating dispersion expressions Drew dispersion curves for modes p = and. .. with radius of 100 Å and 150 Å Constructing an energy expression that interacts between the electron and the longitudinal optical phonon in polarized semiconductor quantum wires ACKNOWLEDGEMENTS