1. Double carbon nanotube antenna as a detector of modulated terahertz radiation V. Semenenko 1, V. Leiman 1, A. Arsenin 1, Yu. Stebunov 1, and V. Ryzhii 2 1 Laboratory of Nanooptics and Femtosecond Electronics, Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia 2 Computational Nanoelectronics Laboratory, University of Aizu, Aizuwakamatsu, Fukushima 965-8580, Japan

2. Outline Introduction Double carbon nanotube antenna Analytical solution Results Conclusion

3. Introduction Some schemes of detectors of modulated terahertz radiation: V. Ryzhii, et al // Appl Phys Lett, 2007 V. Leiman, et al // J. Appl Phys, 2008

4. Introduction Detector scheme on the basis of single-walled carbon nanotubes: Yu. Stebunov et al // Appl. Phys. Ex., 2011 Output signal is the current induced by the variable capacitance: … but there is other scheme:

5. INTRODUCTION Small mass Lower electron collision frequency (in comparison with 2DEG) High contact resistance (about 6.5 kΩ) Main advantages of using metallic SWCNTs as mechanical and plasma resonators: … and main disadvantage:

6. Double carbon nanotube antenna Mathematical model. Plasma resonator: 1) Hydrodynamic model for electron transport in metallic SWCNTs: 2) Maxwell equations, 3) Boundary conditions on the nanotubes surface: Mechanical resonator: dynamic Euler-Bernoully equation for beam deflection: and their ends:

7. Mechanical resonator Dynamic Euler-Bernoully equation: Consider the case when the linear forcecan be represented in the following view: We will find the solution in the form of expansion in a series: whereof orthonormal functions That are eigenfunctions of the homogeneous equation

8. Mechanical resonator Expressions for the eigenmodes:

9. Reducing to the lumped oscillator Set of functionsforms a full basis, so we can expand the function into a series of them: Partial sums of the series are shown in the figure for

10. Lumped mechanical oscillator So, substituting we get: Next we will consider only the oscillation of the general mode (j=1), and the dynamic equation for the lumped oscillator will be:

11. Plasma resonator In the case E z =0 Solution of electrodynamic equations for the nanotubes of infinite length gives: Dispersion equation: Sign “-” corresponds to the anti- symmetric mode, that carries a signal in the double line. In this case i.e. phase velocity doesn’t depend on k. one can obtain an equation for the forced plasma oscillation in the Fourier space: Here is implied that only symmetric mode can be exited in the system, i.e. When

12. Plasma resonator Then it possible to make analytically reverse Fourier transformation of the previous equations system: Boundary conditions: Solution of the system: is the linear charge of the two nanotubes is the electric current in them Because of We consider that the external electric field can be considered as uniform near the system, thus, we can put

13. Lumped plasma resonator Remind the expression for  in  z-space: The frequencies of plasma resonances are For the caseone can put: and introduce a new value

14. Lumped plasma resonator In the vicinity of the general resonance the approximate relation is valid: Graphs are built for the oscillations quality factor

15. Lumped plasma resonator So, in Fourier space we have the relation: or …that corresponds to the following relation in the t-space: and The tz-dependence of ρ will be:

16. Interaction between the resonators 1) Plasma resonator affects the mechanical one: The linear force acting on one nanotube from another: Thus, we obtain: Dynamic equation for the mechanical resonator

17. Interaction between the resonators 1) Mechanical resonator affects the plasma one: Deformation of the plasma resonator causes its eigenfrequency shift. In the case of non-deformed nanotubes we had: When the NTs are deformed, For small oscillation amplitude Wavevector also depends on coordinate:

18. Interaction between the resonators The solution in the case of non-deformed NTs was: To get the solution for the deformed NTs, we should substitute In all the trigonometric functions with: Then the resonant condition will be: or, for general frequencywe will have the relation:

19. Equations for coupled resonators The same structure of the equations as in the case of those describing capacitance transducers:

20. Results Consider the modulated incoming signal: For the small oscillationswe can put Maximumand Amplitude of the nanotubes mechanical oscillation. So, we get: And for the mechanical oscillation:

21. Detection of mech. oscillation Output signal: Thus, If consider this device as a detector of modulated THz radiation, its responsivity will be given as So, for

22. Parametric instability threshold Now consider the monochromatic incoming signal: so, In the assumptionwe can get: and … but actually,

23. Parametric instability threshold So, for the mechanical oscillations we have the following equation: Using the expansion of F into the series of the  degrees, we get:

24. Parametric instability threshold So, whenthere is a self-excitation of the mechanical oscillations in the system. so we can estimate For the parameters stated before we obtain: that is equivalent to the incident radiation intensity In comparison, the maximum amplitude of the modulated signal (with modulation depth m=0.1), under which the nanotubes begin touch one another (2x  1 (0)=d) is estimated as: that corresponds to the intensity of

25. Conclusion The model describing combined plasma and mechanical oscillations in the system of the two parallel metallic carbon nanotubes is developed. Proposed scheme detecting modulated THz radiation featured a remarkably high responsivity (about 10 6 V/W). For the self-excitation of the mechanical modes in the system, quite a high preamplification (by a factor about 10-100) of the incoming signal is needed.

26. Thank you very much for your attention