Ultrasensitive force detection with a nanotube mechanical resonator

Abstract

Since the advent of atomic force microscopy [1], mechanical resonators have been used to study a wide variety of phenomena, such as the dynamics of individual electron spins [2], persistent currents in normal metal rings [3], and the Casimir force [4,5].Key to these experiments is the ability to measure weak forces. Here, we report on force sensing experiments with a sensitivity of 12 zN/ √ Hz at a temperature of 1.2 K using a resonator made of a carbon nanotube. An ultra-sensitive method based on cross-correlated electrical noise measurements, in combination with parametric downconversion, is used to detect the low-amplitude vibrations of the nanotube induced by weak forces. The force sensitivity is quantified by applying a known capacitive force. A promising strategy for measuring lower forces is to employ resonators made of a molecular system, such as a carbon nanotube [14][15][16][17][18]. Nanotube resonators are characterized by an ultra-low mass M, which can be up to seven orders of magnitude lower than that of the ultra-soft cantilevers mentioned above [7], whereas their quality factor Q can be high [19] and their spring constant k 0 low. This has a great potential for generating an outstanding force sensitivity, whose classical limit is given byHere k B T is the thermal energy and γ the mechanical resistance [7]. This limit is set To efficiently convert weak forces into sizable displacements, we design nanotube resonators endowed with spring constants as low as ∼ 10 µN/m. This is achieved by fabri-2 cating the longest possible single-wall nanotube resonators. The fabrication process starts with the growth of nanotubes by chemical vapor deposition onto a doped silicon substrate coated with silicon oxide. Using atomic force microscopy (AFM), we select nanotubes that are straight over a distance of several micrometers, so that they do not touch the underlying substrate once they are released [21]. We use electron-beam lithography to pattern a source and a drain electrode that electrically contact and mechanically clamp the nanotube. We suspend the nanotube using hydrofluoric acid and a critical point dryer. Figure 1a shows a nanotube resonator that is 4 µm long. We characterize its resonant frequencies by driving it electrostatically and using a mixing detection method [18,22]. The lowest resonant frequency is 4.2 MHz (Fig. 1c). This gives a spring constant of 7 µN/m using an effective mass of 10 −20 kg, estimated from the size of the nanotube measured by AFM (supplementary information). This spring constant is comparable to that of the softest cantilevers fabricated so far [6]. When changing the gate voltage V DC g applied to the silicon substrate, the resonant frequency splits into two branches (Fig. 1c). These two branches correspond to the two fundamental modes; they vibrate in perpendicular directions (inset to Fig. 1c).We have developed an ultrasensitive detection method based on parametric downconversion, which (i) employs a cross-correlation measurement scheme to reduce the electrical noise ...

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Single-crystal diamond nanomechanical resonators with quality factors exceeding one million

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“…The subsequent emergence of nanomechanical oscillators and evolutions in readout techniques [2][3][4] lead to impressive improvements in force sensitivity [5], enabling detection of collective spin dynamics [6][7][8], single electron spin [9], mass sensing of atoms [10,11] or inertial sensing [12]. Attractive perspectives arise too when nanoresonators are hybridized to single quantum systems, such as molecular magnets [13], spin or solid states qubits [14][15][16][17].…”