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Asymptotic and Numerical Study

• f Resonant Tunneling in Quantum Waveguides

with Several Equal Resonators

Oleg Sarafanov Differential Operators on Graphs and Waveguides Graz, Austria, February 25 – March 1, 2019

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

Statement of the problem

Figure: Waveguide G♣εq

The wave function u of a free electron of energy k2 satisfies the boundary value problem ✁ ✁∆ ✁ k2✠ u♣x, y; εq ✏ 0, in G♣εq; u♣x, y; εq ✏ 0, on ❇G♣εq.

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

Radiation conditions

Auxiliary boundary value problem in the cross-section of the strip ✁ ✁ ❇2 ❇y2 ✁ λ2✠ Ψ♣yq ✏ 0, Ψ♣✁L④2q ✏ Ψ♣L④2q ✏ 0. ñ λ2

q ✏

✂πq L ✡2 , q ✏ 1, 2, . . . . We assume that λ2

1 ➔ k2 ➔ λ2 2,

and set the radiation conditions u♣x, y; εq ✏ ★ eiνxΨ1♣yq S11♣k; εqe✁iνxΨ1♣yq O♣eδxq, x Ñ ✁✽; S12♣k; εqeiνxΨ1♣yq O♣e✁δxq, x Ñ ✽; δ →0 , ν ✏ ❜ k2 ✁ λ2

1 ,

Ψ1♣yq ✏ ❝ 1 Lν cos πy L .

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

Resonant tunneling

Reflection coefficient and transition coefficient R♣k; εq ✏|S11♣k; εq|2 , T♣k; εq ✏|S12♣k; εq|2 . We have R♣k; εq T♣k; εq ✏ 1. We are interested in

1 The asymptotics of resonant energies k2

r ♣εq as ε Ñ 0;

2 The heights T♣kr, εq of resonant peaks; 3 The width Υ♣εq of resonant peaks at half-height. Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

Single resonator — reference

Figure: Waveguide G♣εq with a single resonator

• L. Baskin, P. Neittaanm¨

aki, B. Plamenevskii, and O. Sarafanov. Resonant Tunneling: Quantum Waveguides of Variable Cross-Section, Asymptotics, Numerics, and Applications // Lecture Notes on Numerical Methods in Engineering and Sciences, Springer, 2015, 275 pp.

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

Single resonator — asymptotics

The results obtained: k2

r ♣εq ✏ k2 e ✁ Qε2π④ω O♣ε2π④ω2✁δq,

T♣k; εq ✏ 1 1 P2 ✄ k2 ✁ k2

r

ε4π④ω ☛2 ✁ 1 O♣ε2✁δq ✠ , Υ♣εq ✏ 2 P ε4π④ω.

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

n equal resonators — resonant energies

Figure: Waveguide G♣εq with n equal resonators

The resonant energies k2

r,j♣εq ✏ k2 e ✁ Qjε2π④ω O♣ε4π④ωq,

j ✏ 1, . . . , n; Qj ✏ A 2B cos πj n 1.

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

n equal resonators — peaks

The transition coefficient and the widths of peaks T♣k; εq ✏ 1 1 P2♣nq ✄➧n

j✏1♣k2 ✁ k2 r,jq

ε2♣n1qπ④ω ☛2 ✁ 1 O♣ε4π④3ωq ✠ ; Υj♣εq ✏ 4 ♣n 1qP♣nqBn✁1 sin2 ✂ πj n 1 ✡ ε4π④ω.

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

Numerics for one resonator

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

ǫ

25 26 27 28 29 30 31

k2

res num asym

Figure: Resonant energy

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

ǫ

0.5 1 1.5 2 2.5 3

Υ

Figure: Width of a resonant peak

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

Numerics for 10 resonators

• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 0.03 0.04

k2-k2

res,middle

0.2 0.4 0.6 0.8 1

T1

Figure: Resonant peaks for ε ✏ 0.1

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

k2-k2

res,middle

0.2 0.4 0.6 0.8 1

T1

Figure: Resonant peaks for ε ✏ 0.3

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum

Thank you!

Oleg Sarafanov Asymptotic and Numerical Study of Resonant Tunneling in Quantum