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Rashba coupling in three-electron-quantum dot under cylindrical symmetry: An exact solution
Annals of Physics 326 (2011) 2957–2962
Contents lists available at SciVerse ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
Rashba coupling in three-electron-quantum dot under cylindrical symmetry: An exact solution H. Hassanabadi a,∗ , H. Rahimov b , S. Zarrinkamar c a
Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316 Shahrood, Iran
b
Computer Engineering Department, Shahrood University of Technology, Shahrood, Iran
c
Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran
article
info
Article history: Received 2 June 2011 Accepted 30 July 2011 Available online 23 August 2011 Keywords: Three-electron-quantum dot Rashba spin–orbit interaction Schrödinger equation
abstract The application of quantum dots in advanced technology goes beyond doubt. Here, based on an analytical methodology, investigate a three-electron-quantum dot in the presence of Rashba spin–orbit interaction under cylindrical symmetry. Both eigenvalues and eigenfunctions of the system are reported and the problem is numerically discussed. © 2011 Elsevier Inc. All rights reserved.
1. Introduction The quantum dots have been becoming an appealing topic in the annals of few body systems for their increasing applications in science and technology as well as their attractive and complicated mathematical nature. In such systems, the effect of spin–orbit interaction (SOI) in the spectrum is definitely important. In the jargon, we frequently face the Dresselhaus and Rashba terms which are respectively due to the electric field produced by the bulk inversion asymmetry of the material and the structural asymmetry of the hetero structure. There is now no doubt about the dependence of optical and electrical properties of confined electrons in quantum dots, wells and wires on the Rashba spin–orbit coupling [1–3] and the experimental data do verify this claim [4–9]. Some interesting features of such systems as well as related topics and applications can be found in Refs. [8–15].
∗
Corresponding author. Tel.: +98 273 3395270; fax: +98 273 3395270. E-mail address: [email protected] (H. Hassanabadi).
0003-4916/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2011.07.011
2958
H. Hassanabadi et al. / Annals of Physics 326 (2011) 2957–2962
2. Rashba spin–orbit interaction The strength of Rashba and Dresselhaus interactions depends on the characteristics of the material and can be controlled by an external electric field as well. In the absence of SOI, the effect of confinement is easily explained by the well-known Fock–Darwin basis (harmonic potential) or an extension of the Landau problem eigenfunctions to the disk geometry (hard-wall). Here, we intend to work on the Rashba term which for a single electron has the form (j)
HS–O = αR (σx(j) py(j) − σy(j) p(xj) ),
(1)
and the standard Pauli matrices are
σx =
0 1
1 , 0
σy =
0 i
−i
.
0
(2)
By introducing Jacobi coordinates
⃗= R
⃗r1 + ⃗r2 + ⃗r3 3
,
ρ⃗ =
⃗r1 − ⃗r2 √ ,
λ⃗ =
2
⃗r1 + ⃗r2 − 2⃗r3 , √
(3)
6
the total spin–orbit for three-electron quantum dot takes the form HS–O =
3 −
(j)
HSO = αR
3 − (σx(j) p(yj) − σy(j) p(xj) )
j =1
j =1
3 − ∂ ∂ i + ∂ xj ∂ yj j =1 , 0
=
0 3 i − −i ∂ + ∂ ∂ xj ∂ yj j =1
h¯ αR
i=
√ −1.
(4)
On the other hand, using Eq. (3) we simply have 3 − ∂ ∂ = , ∂ x ∂ Rx j j=1
3 − ∂ ∂ = . ∂ y ∂ Ry j j =1
(5)
Following the above transformations, we now arrive at
3
HS–O =
−
(j)
HSO =
h¯ αR i
j =1
0
i
∂ ∂ −i + ∂ Rx ∂ Ry
∂ ∂ + ∂ Rx ∂ Ry ,
(6)
0
which means the SOI only depends on the center of mass component. 2.1. The relative part The Hamiltonian Htot of the three-electron-quantum dot in two dimensions with the Coulomb potential and Rashba SOI is Htot =
3 −
j,l=1 j
p2j 2m∗e
+
e2
(j)
+ HSO κ ⃗rj − ⃗rl
+ VCon ,
(7)
where the confinement potential VCon is considered as
VCon =
0
∞
ρ < ρ0 ρ > ρ0 .
(8)
H. Hassanabadi et al. / Annals of Physics 326 (2011) 2957–2962
2959
For the kinetic energy part, using Eq. (3) we find 3 − p2j
2m∗
p2c .m.
=
2M ∗
e
j =1
+
e
p2ρ
+
2mρ
p2λ 2mλ
,
(9)
where Me∗ = 3m∗e and mρ = mλ = m∗e . Let us now use the hyperspherical coordinates including the hyperradius x and the hyperangle ς respectively defined via [4]:
2 x = ρ + λ2 ,
ζ = Arc tan
ρ λ
.
(10)
Consequently, p2ρ 2mρ
p2λ
+
=−
2mλ
d2
1 2m∗e
dx2
+
3d
γ (γ + 2)
−
dx
x2
,
(11)
and 3 − i,j=1 i
e2
e2
√
= κx κ ⃗ri − ⃗rj
1 + α2
2(1 + α 2 )
+
√
2α
√
3+α
+
2(1 + α 2 )
√
3−α
c
= , x
(12)
|λ⃗ |
where α = |⃗ρ | and c=
e2
√
1 + α2
√
2α
k
+
2(1 + α 2 )
√
3+α
+
2(1 + α 2 )
√
3−α
.
(13)
To summarize the above equations, the Hamiltonian now appears as
Htot =
p2c .m. 2Me∗
+
h¯ αR
0
i −i ∂ + ∂ ∂ Rx ∂ Ry
∂ ∂ i + 2 ∂ Rx ∂ Ry + VCon + px + c , 2m∗e x 0
(14)
and the total Hamiltonian Htot is separated in terms of the center of mass and relative coordinate parts, i.e. Htot = H (Rc .m. ) + H (x),
(15)
where H (Rc .m. ) contains VCon . Eq. (15) alternatively can be stated as Em,n,γ = Em,c .m. + En,γ ,
(16)
where m, n and γ denote the quantum numbers. Therefore, we show the wavefunctions by
Ψn,γ ,m (x, Rc .m. ) = Qn,γ (x)ψm (Rc .m. ),
(17)
where
ψm (Rc .m. ) =
u(Rc .m. )eimϕ w(Rc .m. )ei(m+1)ϕ
,
(18)
in which the radial wavefunctions u(Rc .m. ) and w(Rc .m. ) should be regular at the origin. Inserting Eq. (18) in Eq. (14), the hyperradius part is written as
[ 2 ] h¯ 2 d 3 d γ (γ + 2) c − ∗ + − + Qn,γ (x) = En,γ Qn,γ (x), 2 2 2me
dx
x dx
x
x
(19)
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H. Hassanabadi et al. / Annals of Physics 326 (2011) 2957–2962
whose eigenvalues and eigenfunction are
En,γ =
m∗e
−m∗e c 2
e2 k
[√
=− 3 2
2 n+γ +
2 n+γ +
√
−2m∗e En,γ x 2γ +2
Qn,γ (x) = Nn,γ x e
2(1+α 2 ) √ 3+α
+
2α
2
γ − 1h¯
Ln
√
√ 1+α 2
√
h¯
−2m∗ E
e n,γ x
]2 ,
3 2 2
2
2(1+α 2 ) √ 3−α
+
,
(20)
(21)
respectively, and Nn,γ is the normalization constant and Lkn (x) shows the well-known Laguerre polynomial. 2.2. The center of mass part Based on the above arguments, the center of mass equation H (Rc .m. ) is written as
0 − h2 ∂ 2 2 ∂ αR h¯ ¯ + + 2M ∗ ∂ R2x ∂ R2y i −i ∂ + ∂ e ∂ Rx ∂ Ry
i
∂ ∂ + ∂ Rx ∂ Ry ψm (R) 0
= (Em,c .m. )ψm (R). By choosing Rc .m. = R2x + R2y and introducing other dimensionless parameters according to εm =
2Me∗ R20,c .m. Ec .m. h¯
2
,
α=
2Me∗ R0,c .m. αR h¯
,
R=
Rc .m. R0,c .m.
,
(22)
(23)
Eq. (22) yields [14,15] R
2d
R2
2
u
dR2 d2 w dR2
+R
du dR
+R
+ (k k R − m )u − (k − k )R + − 2
dw dR
2
+
−
2
dw dR
+
+ (k+ k− R2 − (m + 1)2 )w + (k+ − k− )R2
m+1 R
du dR
w = 0,
−
m R
u
= 0,
(24a)
(24b)
where k± (εm , α) = εm + α4 ± α2 and we can find that k+ k− = εm and k+ − k− = α . On the other hand, the quite known properties of Bessel functions give [14,15] 2
u(m, εm , α, R) = Af (m, εm , α, R) + Bg (m, εm , α, R),
(25a)
w(m + 1, εm , α, R) = Ag (m + 1, εm , α, R) + Bf (m + 1, εm , α, R),
(25b)
where
1 Jm (k− R) + Jm (k+ R) , 2 1 g (m, εm , α, R) = Jm (k− R) − Jm (k+ R) . 2 Now, the continuity conditions at Rc .m. = R0,c .m. or R = 1 imply f (m, εm , α, R) =
(26a) (26b)
u(m, εm , α, 1) = 0,
(27a)
w(m, εm , α, 1) = 0.
(27b)
From Eq. (27b), A can be written in the form A = −B
f (m + 1, εm , α, 1) g (m + 1, εm , α, 1)
.
(28)
H. Hassanabadi et al. / Annals of Physics 326 (2011) 2957–2962
2961
Fig. 1. Upper and lower components of wavefunction vs. R.
By substitution of relation (28) in Eq. (27a) we find
−f (m + 1, εm , α, 1)f (m, εm , α, 1) + g (m, εm , α, 1)g (m + 1, εm , α, 1) = 0, g (m + 1, εm , α, 1)
(29)
or equivalently Jm+1 (k− ) − Jm+1 (k+ )
Jm (k− ) − Jm (k+ )
− Jm (k− ) + Jm (k+ ) Jm+1 (k− ) + Jm+1 (k+ ) = 0,
(30)
and the energy eigenvalues Em,c .m. of the system are found from the above equation. As a result, the function u(m, εm , α, R) and w(m + 1, εm , α, R) are
(Jm+1 (k− ) + Jm+1 (k+ )) Jm (k− R) + Jm (k+ R) − + (Jm+1 (k ) − Jm+1 (k )) − + + Jm (k R) − Jm (k R) ,
u(m, εm , α, R) = B −
(31a)
(Jm+1 (k− ) + Jm+1 (k+ )) Jm+1 (k− R) − Jm+1 (k+ R) (Jm+1 (k− ) − Jm+1 (k+ )) − + + Jm+1 (k R) + Jm+1 (k R) .
w(m + 1, εm , α, 1) = B −
(31b)
In Fig. 1, we have plotted the upper and lower components vs. R for difference ms and Table 1 reports numerical values of different states. 3. Conclusion The numerical analysis is time-consuming, less familiar and more cumbersome in comparison with the analytical techniques of mathematical physics. Here, using the properties of well-known special functions of mathematical physics as well as Jacobi coordinates and hyperspherical coordinates, we have reported exact analytical solutions of the problem under cylindrical symmetry and Rashba spin orbit interaction. Our study may provide an acceptable basis for related fields. The results reveal that
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H. Hassanabadi et al. / Annals of Physics 326 (2011) 2957–2962
Table 1 Eigenvalues for different states.
α
State |n, γ , m⟩
State |n, γ , m⟩
Energy
State |n, γ , m⟩
Energy
|1, 0, 1⟩
2.65239 2.33744 −2.05656 −9.71013
|1, 0, 2⟩
−2.2416 −2.2416 −2.90953
0.784058 0.621638 1.65989 −0.753429
|2, 0, 1⟩
3.75032 3.43537 −0.95863 −8.6122
|2, 0, 2⟩
−1.14367 −1.14367 −1.81161
1.23588 1.07346 2.11171 −0.301608
|2, 1, 1⟩
4.20214 3.88719 −0.506809 −8.16038
|2, 1, 2⟩
−0.691851 −0.691851 −1.35979
Energy
−0.313867 −0.476287
0 1 5 10
|1, 0, 0⟩
0 1 5 10
|2, 0, 0⟩
0 1 5 10
|2, 1, 0⟩
0 1 5 10
|1, 0, 3⟩
−2.2416 −2.2416 −2.2416
0 1 5 10
|2, 0, 3⟩
−1.14367 −1.14367 −1.14367
0 1 5 10
|2, 1, 3⟩
−0.691851 −0.691851 −0.691851
0.561963
−1.85135
11.3272
6.54994
7.64787
8.09969
38.5677
|1, 0, 4⟩
−2.2416
|2, 0, 4⟩
−1.14367
|2, 1, 4⟩
−0.691851
12.4251
7.50471 6.39747
|1, 0, 5⟩
23.4047 22.5045 −2.24158 −2.43698
|2, 0, 5⟩
24.5026 23.6024 −1.14365 −1.33906
|2, 1, 5⟩
24.9544 24.0542 −0.69183 −0.887237
39.6656
12.8769
8.60264 7.4954 40.1174 9.05446 7.94722
Rashba interaction appears solely in the center of mass section and, the higher the corresponding quantum number m is, the higher the energy of the system is. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
E.I. Rashba, Fiz. Tverd. Tela (Leningrad) 2 (1960) 1224. [Sov. Phys. Solid State 2, 1109 (1960)]. Yu.A. Bychkov, E.I. Rashba, J. Phys. C 17 (1984) 6039. J. Nitta, et al., Phys. Rev. Lett. 78 (1997) 1335. J.B. Miller, et al., Phys. Rev. Lett. 90 (2003) 076807. B. Basu, B. Roy, European J. Phys. 30 (2009) 955–963. F.E. Meijer, et al., Phys. Rev. B 66 (2002) 033107. J. Splettstoesser, et al., Phys. Rev. B 68 (2003) 165341. J. Nitta, et al., Appl. Phys. Lett. 75 (1999) 695. A.G. Aronov, Y.B. Lyanda-Geller, Phys. Rev. Lett. 70 (1993) 343. H. Hassanabadi, M. Hamzavi, S. Zarrinkamar, A.A. Rajabi, Few-Body Syst. 48 (2010) 53–58. T. Koga, et al., Phys. Rev. B 70 (2004) 161302(R). A.F. Morpurgo, et al., Phys. Rev. Lett. 80 (1998) 1050. J.-B. Yau, et al., Phys. Rev. Lett. 88 (2002) 146801. V.V. Kudryashov, Nonlinear Phenom. Complex Syst. 12 (2) (2009) 199–203. E. Tsitsishvili, G.S. Lozano, A.O. Gogolin, Phys. Rev. B 70 (2004) 115316.
Contents lists available at SciVerse ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
Rashba coupling in three-electron-quantum dot under cylindrical symmetry: An exact solution H. Hassanabadi a,∗ , H. Rahimov b , S. Zarrinkamar c a
Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316 Shahrood, Iran
b
Computer Engineering Department, Shahrood University of Technology, Shahrood, Iran
c
Department of Basic Sciences, Garmsar Branch, Islamic Azad University, Garmsar, Iran
article
info
Article history: Received 2 June 2011 Accepted 30 July 2011 Available online 23 August 2011 Keywords: Three-electron-quantum dot Rashba spin–orbit interaction Schrödinger equation
abstract The application of quantum dots in advanced technology goes beyond doubt. Here, based on an analytical methodology, investigate a three-electron-quantum dot in the presence of Rashba spin–orbit interaction under cylindrical symmetry. Both eigenvalues and eigenfunctions of the system are reported and the problem is numerically discussed. © 2011 Elsevier Inc. All rights reserved.
1. Introduction The quantum dots have been becoming an appealing topic in the annals of few body systems for their increasing applications in science and technology as well as their attractive and complicated mathematical nature. In such systems, the effect of spin–orbit interaction (SOI) in the spectrum is definitely important. In the jargon, we frequently face the Dresselhaus and Rashba terms which are respectively due to the electric field produced by the bulk inversion asymmetry of the material and the structural asymmetry of the hetero structure. There is now no doubt about the dependence of optical and electrical properties of confined electrons in quantum dots, wells and wires on the Rashba spin–orbit coupling [1–3] and the experimental data do verify this claim [4–9]. Some interesting features of such systems as well as related topics and applications can be found in Refs. [8–15].
∗
Corresponding author. Tel.: +98 273 3395270; fax: +98 273 3395270. E-mail address: [email protected] (H. Hassanabadi).
0003-4916/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2011.07.011
2958
H. Hassanabadi et al. / Annals of Physics 326 (2011) 2957–2962
2. Rashba spin–orbit interaction The strength of Rashba and Dresselhaus interactions depends on the characteristics of the material and can be controlled by an external electric field as well. In the absence of SOI, the effect of confinement is easily explained by the well-known Fock–Darwin basis (harmonic potential) or an extension of the Landau problem eigenfunctions to the disk geometry (hard-wall). Here, we intend to work on the Rashba term which for a single electron has the form (j)
HS–O = αR (σx(j) py(j) − σy(j) p(xj) ),
(1)
and the standard Pauli matrices are
σx =
0 1
1 , 0
σy =
0 i
−i
.
0
(2)
By introducing Jacobi coordinates
⃗= R
⃗r1 + ⃗r2 + ⃗r3 3
,
ρ⃗ =
⃗r1 − ⃗r2 √ ,
λ⃗ =
2
⃗r1 + ⃗r2 − 2⃗r3 , √
(3)
6
the total spin–orbit for three-electron quantum dot takes the form HS–O =
3 −
(j)
HSO = αR
3 − (σx(j) p(yj) − σy(j) p(xj) )
j =1
j =1
3 − ∂ ∂ i + ∂ xj ∂ yj j =1 , 0
=
0 3 i − −i ∂ + ∂ ∂ xj ∂ yj j =1
h¯ αR
i=
√ −1.
(4)
On the other hand, using Eq. (3) we simply have 3 − ∂ ∂ = , ∂ x ∂ Rx j j=1
3 − ∂ ∂ = . ∂ y ∂ Ry j j =1
(5)
Following the above transformations, we now arrive at
3
HS–O =
−
(j)
HSO =
h¯ αR i
j =1
0
i
∂ ∂ −i + ∂ Rx ∂ Ry
∂ ∂ + ∂ Rx ∂ Ry ,
(6)
0
which means the SOI only depends on the center of mass component. 2.1. The relative part The Hamiltonian Htot of the three-electron-quantum dot in two dimensions with the Coulomb potential and Rashba SOI is Htot =
3 −
j,l=1 j
p2j 2m∗e
+
e2
(j)
+ HSO κ ⃗rj − ⃗rl
+ VCon ,
(7)
where the confinement potential VCon is considered as
VCon =
0
∞
ρ < ρ0 ρ > ρ0 .
(8)
H. Hassanabadi et al. / Annals of Physics 326 (2011) 2957–2962
2959
For the kinetic energy part, using Eq. (3) we find 3 − p2j
2m∗
p2c .m.
=
2M ∗
e
j =1
+
e
p2ρ
+
2mρ
p2λ 2mλ
,
(9)
where Me∗ = 3m∗e and mρ = mλ = m∗e . Let us now use the hyperspherical coordinates including the hyperradius x and the hyperangle ς respectively defined via [4]:
2 x = ρ + λ2 ,
ζ = Arc tan
ρ λ
.
(10)
Consequently, p2ρ 2mρ
p2λ
+
=−
2mλ
d2
1 2m∗e
dx2
+
3d
γ (γ + 2)
−
dx
x2
,
(11)
and 3 − i,j=1 i
e2
e2
√
= κx κ ⃗ri − ⃗rj
1 + α2
2(1 + α 2 )
+
√
2α
√
3+α
+
2(1 + α 2 )
√
3−α
c
= , x
(12)
|λ⃗ |
where α = |⃗ρ | and c=
e2
√
1 + α2
√
2α
k
+
2(1 + α 2 )
√
3+α
+
2(1 + α 2 )
√
3−α
.
(13)
To summarize the above equations, the Hamiltonian now appears as
Htot =
p2c .m. 2Me∗
+
h¯ αR
0
i −i ∂ + ∂ ∂ Rx ∂ Ry
∂ ∂ i + 2 ∂ Rx ∂ Ry + VCon + px + c , 2m∗e x 0
(14)
and the total Hamiltonian Htot is separated in terms of the center of mass and relative coordinate parts, i.e. Htot = H (Rc .m. ) + H (x),
(15)
where H (Rc .m. ) contains VCon . Eq. (15) alternatively can be stated as Em,n,γ = Em,c .m. + En,γ ,
(16)
where m, n and γ denote the quantum numbers. Therefore, we show the wavefunctions by
Ψn,γ ,m (x, Rc .m. ) = Qn,γ (x)ψm (Rc .m. ),
(17)
where
ψm (Rc .m. ) =
u(Rc .m. )eimϕ w(Rc .m. )ei(m+1)ϕ
,
(18)
in which the radial wavefunctions u(Rc .m. ) and w(Rc .m. ) should be regular at the origin. Inserting Eq. (18) in Eq. (14), the hyperradius part is written as
[ 2 ] h¯ 2 d 3 d γ (γ + 2) c − ∗ + − + Qn,γ (x) = En,γ Qn,γ (x), 2 2 2me
dx
x dx
x
x
(19)
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H. Hassanabadi et al. / Annals of Physics 326 (2011) 2957–2962
whose eigenvalues and eigenfunction are
En,γ =
m∗e
−m∗e c 2
e2 k
[√
=− 3 2
2 n+γ +
2 n+γ +
√
−2m∗e En,γ x 2γ +2
Qn,γ (x) = Nn,γ x e
2(1+α 2 ) √ 3+α
+
2α
2
γ − 1h¯
Ln
√
√ 1+α 2
√
h¯
−2m∗ E
e n,γ x
]2 ,
3 2 2
2
2(1+α 2 ) √ 3−α
+
,
(20)
(21)
respectively, and Nn,γ is the normalization constant and Lkn (x) shows the well-known Laguerre polynomial. 2.2. The center of mass part Based on the above arguments, the center of mass equation H (Rc .m. ) is written as
0 − h2 ∂ 2 2 ∂ αR h¯ ¯ + + 2M ∗ ∂ R2x ∂ R2y i −i ∂ + ∂ e ∂ Rx ∂ Ry
i
∂ ∂ + ∂ Rx ∂ Ry ψm (R) 0
= (Em,c .m. )ψm (R). By choosing Rc .m. = R2x + R2y and introducing other dimensionless parameters according to εm =
2Me∗ R20,c .m. Ec .m. h¯
2
,
α=
2Me∗ R0,c .m. αR h¯
,
R=
Rc .m. R0,c .m.
,
(22)
(23)
Eq. (22) yields [14,15] R
2d
R2
2
u
dR2 d2 w dR2
+R
du dR
+R
+ (k k R − m )u − (k − k )R + − 2
dw dR
2
+
−
2
dw dR
+
+ (k+ k− R2 − (m + 1)2 )w + (k+ − k− )R2
m+1 R
du dR
w = 0,
−
m R
u
= 0,
(24a)
(24b)
where k± (εm , α) = εm + α4 ± α2 and we can find that k+ k− = εm and k+ − k− = α . On the other hand, the quite known properties of Bessel functions give [14,15] 2
u(m, εm , α, R) = Af (m, εm , α, R) + Bg (m, εm , α, R),
(25a)
w(m + 1, εm , α, R) = Ag (m + 1, εm , α, R) + Bf (m + 1, εm , α, R),
(25b)
where
1 Jm (k− R) + Jm (k+ R) , 2 1 g (m, εm , α, R) = Jm (k− R) − Jm (k+ R) . 2 Now, the continuity conditions at Rc .m. = R0,c .m. or R = 1 imply f (m, εm , α, R) =
(26a) (26b)
u(m, εm , α, 1) = 0,
(27a)
w(m, εm , α, 1) = 0.
(27b)
From Eq. (27b), A can be written in the form A = −B
f (m + 1, εm , α, 1) g (m + 1, εm , α, 1)
.
(28)
H. Hassanabadi et al. / Annals of Physics 326 (2011) 2957–2962
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Fig. 1. Upper and lower components of wavefunction vs. R.
By substitution of relation (28) in Eq. (27a) we find
−f (m + 1, εm , α, 1)f (m, εm , α, 1) + g (m, εm , α, 1)g (m + 1, εm , α, 1) = 0, g (m + 1, εm , α, 1)
(29)
or equivalently Jm+1 (k− ) − Jm+1 (k+ )
Jm (k− ) − Jm (k+ )
− Jm (k− ) + Jm (k+ ) Jm+1 (k− ) + Jm+1 (k+ ) = 0,
(30)
and the energy eigenvalues Em,c .m. of the system are found from the above equation. As a result, the function u(m, εm , α, R) and w(m + 1, εm , α, R) are
(Jm+1 (k− ) + Jm+1 (k+ )) Jm (k− R) + Jm (k+ R) − + (Jm+1 (k ) − Jm+1 (k )) − + + Jm (k R) − Jm (k R) ,
u(m, εm , α, R) = B −
(31a)
(Jm+1 (k− ) + Jm+1 (k+ )) Jm+1 (k− R) − Jm+1 (k+ R) (Jm+1 (k− ) − Jm+1 (k+ )) − + + Jm+1 (k R) + Jm+1 (k R) .
w(m + 1, εm , α, 1) = B −
(31b)
In Fig. 1, we have plotted the upper and lower components vs. R for difference ms and Table 1 reports numerical values of different states. 3. Conclusion The numerical analysis is time-consuming, less familiar and more cumbersome in comparison with the analytical techniques of mathematical physics. Here, using the properties of well-known special functions of mathematical physics as well as Jacobi coordinates and hyperspherical coordinates, we have reported exact analytical solutions of the problem under cylindrical symmetry and Rashba spin orbit interaction. Our study may provide an acceptable basis for related fields. The results reveal that
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Table 1 Eigenvalues for different states.
α
State |n, γ , m⟩
State |n, γ , m⟩
Energy
State |n, γ , m⟩
Energy
|1, 0, 1⟩
2.65239 2.33744 −2.05656 −9.71013
|1, 0, 2⟩
−2.2416 −2.2416 −2.90953
0.784058 0.621638 1.65989 −0.753429
|2, 0, 1⟩
3.75032 3.43537 −0.95863 −8.6122
|2, 0, 2⟩
−1.14367 −1.14367 −1.81161
1.23588 1.07346 2.11171 −0.301608
|2, 1, 1⟩
4.20214 3.88719 −0.506809 −8.16038
|2, 1, 2⟩
−0.691851 −0.691851 −1.35979
Energy
−0.313867 −0.476287
0 1 5 10
|1, 0, 0⟩
0 1 5 10
|2, 0, 0⟩
0 1 5 10
|2, 1, 0⟩
0 1 5 10
|1, 0, 3⟩
−2.2416 −2.2416 −2.2416
0 1 5 10
|2, 0, 3⟩
−1.14367 −1.14367 −1.14367
0 1 5 10
|2, 1, 3⟩
−0.691851 −0.691851 −0.691851
0.561963
−1.85135
11.3272
6.54994
7.64787
8.09969
38.5677
|1, 0, 4⟩
−2.2416
|2, 0, 4⟩
−1.14367
|2, 1, 4⟩
−0.691851
12.4251
7.50471 6.39747
|1, 0, 5⟩
23.4047 22.5045 −2.24158 −2.43698
|2, 0, 5⟩
24.5026 23.6024 −1.14365 −1.33906
|2, 1, 5⟩
24.9544 24.0542 −0.69183 −0.887237
39.6656
12.8769
8.60264 7.4954 40.1174 9.05446 7.94722
Rashba interaction appears solely in the center of mass section and, the higher the corresponding quantum number m is, the higher the energy of the system is. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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