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Quasi-one-dimensional ballistic ring in the field of circularly polarized electromagnetic wave E.M. Epshtein1 , E.G. Fedorov2 , G.M. Shmelev3 1
Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Fryazino, Russia; e-mail:
[email protected] 2
Volgograd State University of Architecture and Civil Engineering, Volgograd, Russia; e-mail:
[email protected] 3
Volgograd State Pedagogical University, Volgograd, Russia; e-mail:
[email protected]
Dynamics is studied of an electron in a quasi-one-dimensional ballistic ring under circularly polarized electromagnetic field propagating along the normal to the ring plane. The average emission intensity from the ring is calculated. The value and direction of the electron average angular velocity in the ring depend on the incident wave parameters. It is found that the ring average dipole moment can remain constant under certain conditions. Possibility is shown of higher harmonics enhancement in the ring radiation spectrum.
Studying properties of quasi-one-dimensional rings is one of the most promising directions in physics of low-dimensional structures (e.g., [1, 2]). Incidentally, quantum phenomena were considered preferably. Meanwhile, quasi-one-dimensional rings have interesting classical electrodynamic properties, too [3 – 9]. In present work, we investigate electromagnetic response of a quasi-one-dimensional ring in a circularly polarized (CP) electromagnetic field propagating along the normal to the ring plane. Consider a plane ring with width small compared to its radius R. Such a ring is a quantum well between two concentric potential barriers with quantum confinement along the radius. We assume that the electron mean free path is large in comparison to 2πR, so that the electron executes ballistic classic motion along the ring circumference. Before the CP wave turning on at t = 0, the electron with energy W moves along the ring circumference with constant angular
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velocity. Let a CP wave is incident on the ring with the wavelength large compared to the ring diameter. In dipole approximation, only the electric field of the wave E = E0 {cos (ωt + β ), sin (ω t + β )}
(1)
acts on the electrons. The collisionless motion of the electrons interacting with the CP wave is described by equation
&ϕ& + Ω 2 sin (ϕ − ωt − α ) = 0 ,
(2)
where ϕ = ϕ(t ) is angular coordinate counted off the Ox axis, α = β ± π ,
Ω2 =
e E0 mR
,
(3)
e and m are electron charge and effective mass, respectively. The solution of Eq. (2) is
arcsin [k sn (Ψ1 , k )], k < 1, ϕ(t ) = ωt + α + 2arcsin [tanh (Ψ2 )], k = 1, −1 arcsin sn Ψ3 , k , k > 1,
[ (
(4)
)]
where
ϕ(0) − α Ψ1 = ξΩt + F arcsin k −1 sin , k 2
( k < 1),
ϕ(0 ) − α Ψ2 = ξΩt + F , 1 , 2 ϕ(0) − α −1 Ψ3 = kξΩt + F ,k 2
(5)
(6)
(k > 1),
& (0) − ω) , ξ = sgn (ϕ
(7) (8)
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[ϕ& (0) − ω]2 1 2 k= + [ 1 − cos ( ϕ ( 0 ) − α ) ] , 2 2 4Ω
(9)
12 ϕ& (0 ) = ± R −1 (2W m ) is angular velocity of the electron with energy W at t = 0, F(x, q) is
incomplete elliptic integral of the first kind, sn(x, q) is elliptic sine. Let the initial conditions ϕ(0) , ϕ& (0) be the same for all the N electrons in the ring. The dipole moment with respect to the ring center and the radiation intensity are
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P = NeR{cos ϕ, sin ϕ}
(10)
and I (t ) =
(
)
2 && 2 (P ) = 23 (NeR )2 ϕ& 4 + &ϕ& 2 , 3 3c 3c
(11)
respectively [10]. For an electron in the ring under CP wave acting, we have ϕ& 4 = ω 4 + 8ξω3ΩG + 24ω 2 Ω 2 G 2 + 32ξω Ω3 G 3 + 16Ω 4 G 4 ,
(12)
k cn(Ψ1 , k ), k < 1, G = sech Ψ2 , k −1 , k = 1, −1 k dn Ψ3 , k , k > 1,
(13)
k 2 sn 2 (Ψ1 , k ) dn 2 (Ψ1 , k ), k < 1, && 2 = 4Ω 4 sinh 2 (Ψ2 ) sech4 (Ψ2 ), k = 1, ϕ 2 −1 2 −1 sn Ψ3 , k cn Ψ3 , k , k > 1,
(14)
where
( (
) )
(
) (
)
cn(x, q) is elliptic cosine, dn(x, q) is delta amplitude. The radiation fundamental frequency defined by Eq. (11) is equal to
1 2 K (k ) , k < 1, ω0 = Ωπ0, k = 1, k , k > 1, K k −1
(15)
( )
where K(q) is complete elliptic integral of the first kind. The dependence of ω0 /Ω on q is shown in Fig. 1.
Fig. 1. The dependence of ω0 /Ω on q.
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It follows from Eq. (11) that the average radiation intensity from the ring is
(
)
ω4 + 2k 2 Ω 2 6ω2 + Ω 2 + 6Ω 4 k 4 , k > 1,
where the angular brackets mean averaging over period T = 2π ω 0 , k being defined by Eq. (9). At arbitrary k, the average intensity I can be found numerically from Eq. (11), T
I = T −1 ∫ I (t )dt .
(17)
0
If the following conditions fulfil
ω = ϕ& (0 ), β = ϕ (0), ϕ (0) ± π,
(18)
then E field is collinear to P vector at any instant, so that the electron moves along the ring circumference with constant angular velocity ϕ& = ϕ& (0) = ω . At tha t case, the radiation intensity is [10] I=
2 (NeR )2 ω4 . 3 3c
(19)
2 4 −3 Note that the radiation intensity is I 0 = (2 3)c (NeR) ϕ& (0) up to t = 0 instant.
According with Eq. (4), the electron angular velocity averaged over the period T = 2π ω 0 is
ω, k ≤ 1, ϕ& = k ω + ξπ Ω , k > 1. K k −1
( )
(20)
Under ϕ&
2πω−1 , 2πω0−1 (ν is electron collision frequency). During t ∗ time, the average
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dipole moment P remains almost constant under CP wave acting. The latter fact is confirmed with numerical solution of the equation of motion (2). Let us make numerical estimates. At R = 10–5 cm, m = 0.1me , W = 10–3 eV, we have ϕ& (0) = ±5.9 ⋅10 11 c −1 . The dependence of I I 0 ratio on k = k (ω) found numerically at E0 =
3⋅103 V/cm, β = 5π/4, ϕ(0) = π/4 is shown in Fig. 2.
Fig. 2. I I 0 as a function of k = k (ω ) with E0 = const .
The dependence I
I 0 on k = k (E 0 ) at ω = 1 ⋅10 12 c −1 and the same values of ϕ(0), ϕ& (0)
and β is shown in Fig. 3.
Fig.3. I I 0 as a function of k = k (E 0 ) with ω = const .
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The radiation spectrum at parameters indicated is shown in Fig. 4.
Fig. 4. The ring radiation spectrum ( γ i = i ω0 ϕ& ( 0 ) ). Since I(t) function is periodical at given parameters ( k ≈ 0.089 ≠ 1 ), the spectrum is presented by a correspondence
iω0 → ai2 + bi2 ≡ wi ,
i = 1, 2, ...,
(22)
where ai and bi are coefficients in the I(t) function expansion into Fourier series ∞
I (t ) = I + I 0 ∑ [ai cos(i ω0 t ) + bi sin (iω0 t )].
(23)
i =1
The condition (21) fulfils, for example, at E0 = 3⋅102 V/cm, ω = 9.4⋅1011 c–1 , β = 5 π 4 ,
ϕ (0 ) = π 4 , ϕ& (0) = −5.9 ⋅1011c −1 . In that case, k ≈ 1.06 , ϕ& ≈ 3.4 ⋅ 107 c −1 , 2π/t* ≈ 6.3⋅1010 c–1, ω0 ≈ 9.4⋅1011 c–1 , P
(NeR) = { cos ϕ ,
sin ϕ } ≈ {0.655, 0.657} . The dependence of ϕ on τ = t ϕ& (0)
is shown in Fig. 5.
Fig. 5. The dependence of ϕ on τ = t ϕ& (0) .
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The radiation spectrum in that situation is shown in Fig. 6.
Fig. 6. The ring radiation spectrum under (21) condition. Enhancement is seen of higher harmonics in comparison to Fig. 4.
It is seen that the higher harmonics intens ities decrease much more slowly in comparison with the situation corresponding to Fig. 4. Therefore, enhancement of higher harmonics occurs under condition (21) fulfillment. At above -mentioned parameter values and collision frequency ν = 1010 c–1 we have
ϕ& (0) >> 2π ν , as well as ω, ω0 >> 2πν at k ≠ 1 and ω >> 2πν at k = 1. It justifies the collisionless approximation used in description of the electron dynamics in the ring.
The work was supported partly by the Russian Fund of Basic Research (Project No. 0202-16238).
References
[1] A. Lorke, R.J. Luyken, A.O. Govorov et al., Phys. Rev. Lett. 84, 2223 (2000). [2] N.T. Bagraev, A.D. Buravlev, V.K. Ivanov et al., Semiconductors 34, 817 (2000). [3] E.M. Epshtein, G.M. Shmelev, and I.I. Maglevanny, J. Phys. A.: Math. Gen. 33, 6017 (2000). [4] E.M. Epshtein and G.M. Shmelev, Physica Scripta 62, 216 (2000).
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[5] G.M. Shmelev, E.M. Epshtein, and G.A. Syrodoev, Techn. Phys. 45, 1354 (2000). [6] E.M Epshtein, I.I. Maglevanny, and G.M. Shmelev, Phys. Low-Dim. Struct. No. 3/4, 109 (2000). [7] E.M Epshtein, G.M. Shmelev, and I.I. Maglevanny, Phys. Low-Dim. Struct. No. 1/2, 137 (2001). [8] G.M. Shmelev and E.M. Epshtein, Phys. Solid State, 43, 2311 (2001). [9] E.M. Epshtein and G.M. Shmelev, cond-mat/0312025. [10] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (Pergamon, Oxford, 1973).
