Double carbon nanotube antenna as a detector of modulated terahertz radiation

Double carbon nanotube antenna as a detector of modulated terahertz radiation V. Semenenko1, V. Leiman1, A. Arsenin1, Yu. Stebunov1, and V. Ryzhii2 1Laboratory of Nanooptics and Femtosecond Electronics, Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia 2Computational Nanoelectronics Laboratory, University of Aizu, Aizuwakamatsu, Fukushima 965-8580, Japan

Outline • Introduction • Double carbon nanotube antenna • Analytical solution • Results • Conclusion

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

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

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

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

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

Mechanical resonator Expressions for the eigenmodes:

Reducing to the lumped oscillator Set of functions forms 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

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:

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

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

Lumped plasma resonator Remind the expression for r in wz-space: The frequencies of plasma resonances are For the case one can put: and introduce a new value

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

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 ρwill be:

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

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:

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 frequency we will have the relation:

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

Results Amplitude of the nanotubes mechanical oscillation. Consider the modulated incoming signal: For the small oscillations we can put Maximum and So, we get: And for the mechanical oscillation:

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

Parametric instability threshold Now consider the monochromatic incoming signal: so, In the assumption we can get: … but actually, and

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

Parametric instability threshold So, when there 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 (2xy1(0)=d) is estimated as: that corresponds to the intensity of

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 106 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.

Thank you very much for your attention

Double carbon nanotube antenna as a detector of modulated terahertz radiation