## 1. Introduction

Various electrode materials for single-walled carbon nanotube (SWNT) transistors were investigated. Pd electrodes have been used for Ohmic contacts (Javey et al., 2003). Ti electrodes have been used for Schottky contacts for hole conduction (Heinze et al., 2002; Martel et al., 2001; Kamimura & Matsumoto, 2005), and Mg and Ca electrodes for electron conduction (Nosho et al., 2006). Moreover, in the case of sub-μm order channel length at low temperature, SWNT transistors with Ohmic contacts have shown resonant tunneling transistor (RTT) characteristics (Liang et al., 2001), which are also called as Fabry-Perot characteristics, and SWNT transistors with Schottky contacts have shown single-hole transistor (SHT) characteristics (Suzuki et al., 2001), in which the Schottky barriers act as tunneling barriers. Therefore, electrodes materials should be chosen to obtain the desired characteristics.

In this study, we succeeded in fabricating a multifunctional quantum transistor using the particle nature and wave nature of holes in SWNT. This transistor can operate as an RTT and also as an SHT. An RTT is a device that uses the wave nature of hole and an SHT uses the particle nature of hole in the SWNT. Both devices need tunneling barriers at both sides of the quantum island. The RTT needs strong coupling while the SHT needs weak coupling between the quantum island and the electrodes. Usually, these tunneling barriers are made from thin oxide layers, etc. Therefore, the thickness of the tunneling barriers and the coupling strength cannot normally be controlled in a given device. In the present device, however, the Schottky barriers act as the tunneling barriers between the SWNT quantum island and electrodes. Therefore, the thickness of the tunneling barriers and the coupling strength between the SWNT and electrodes can be controlled by the applied gate voltage _{
G
}.

Moreover, a SWNT is a cylindrical material with a diameter of several nanometers. The small diameter makes it possible to detect an electrical field from even a single-charge. Moreover, by observing the relative energy difference between the conducting carrier and the single-charges to be measured, it is possible to define the potential energy of the single-charges to be measured. However, as described in many reports (Kim et al., 2003; Radosavljevic et al., 2002), SWNT electron devices show hysteresis characteristics in gate voltage-drain current characteristics. The hysteresis characteristics are caused by gate-voltage-dependent charge fluctuation, e.g., adsorption of water molecules around a SWNT (Kim et al., 2003), charging into insulator layer around a SWNT (Radosavljevic et al., 2002), and charging into amorphous carbon around a SWNT (Martel et al., 2001) By eliminating these origins of the hysteresis characteristics, the number of fluctuating charges becomes small and a single-charge can be detected by a SWNT multi-functional quantum transistor.

## 2. Materials and methods

### 2.1. Method

We have eliminated the three origins of the hysteresis characteristics of a SWNT field effect transistor mainly pointed out in current reports (Martel et al., 2001; Kim et al., 2003; Radosavljevic et al. 2002; Kamimura & Matsumoto, 2004). To burn out amorphous carbon, we annealed a SWNT at low temperature in oxidizable atmosphere (Kamimura & Matsumoto, et al., 2004). To reduce the number of adsorbed atmosphere molecules, we covered the channel with a silicon dioxide layer. To reduce the number of trap sites in the insulator, we reduced channel length to 73 nm. The SWNT multi-functional quantum transistor fabricated by the process mentioned above shows almost no hysteresis characteristics in the gate voltage rage from -40 to 40 V. Moreover, an abrupt discrete switching of the source-drain current is observed in the electrical measurements of the SWNT multi-functional quantum transistor at 7.3 K. These random telegraph signals (RTS) are attributed to charge fluctuating charge traps near the SWNT multi-functional quantum transistor conduction channel. The current-switching behavior associated with the occupation of individual electron traps is demonstrated and analyzed statistically.

### 2.2. Sample preparations

A schematic of the sample structure is shown in Fig. 1. SWNT was prepared as follows. An n^{+}-Si wafer with a thermally grown 300 nm thick oxide was used as a substrate. Layered Fe/Mo/Si (2 nm/20 nm/40 nm) catalysts were evaporated using an electron-beam evaporator under a vacuum of 10^{-6} Pa. These layered catalysts were patterned on the substrate using the conventional photo-lithography process. SWNT was grown by thermal chemical vapor deposition (CVD) using the mixed gases of hydrogen and argon-bubbled ethanol. After the growth of the SWNT, it was purified by burning out the amorphous carbon around the SWNT in an air atmosphere at a temperature of several hundred degrees Celsius (Kamimura and Matsumoto, 2004). Ti (30 nm) electrodes were deposited on the patterned catalysts as the source and drain, and on the back side of the n^{+}-Si substrate for the gate, using the electron-beam evaporator under a vacuum of 10^{-6} Pa. The distance (* L*) between the source and drain was 73 nm. Thus, a back gate type multi-functional quantum transistor with an SWNT channel was fabricated that had the functions of an RTT and an SHT. The single-charge measurement was carried out with the structure that silicon dioxide layer is on the SWNT channel.

## 3. Results and discussions

### 3.1. Particle-Wave duality

Fig. 2(a) shows the differential conductance d_{
D
}/d_{
D
} characteristic as a function of the _{
G
} at 7.3 K, where the drain voltage was set at 8 mV.

An oscillation characteristic with two oscillation periods is also observed in Fig. 2(a). A large oscillation period of Δ_{
G
} = 1.1 V was at _{
G
} ≥ –16 V and a small oscillation period of Δ_{
G
} = 0.65 V was at _{
G
} ≤ –16 V. Fig. 2(b) shows the d_{
D
}/ d_{
D
} peak on a linear scale at _{
G
} ≥ –16 V and _{
G
} ≤ –16 V. A clear difference in the oscillation period is observed as follows. The peaks of d_{
D
}/ d_{
D
} at _{
G
} ≥ –16 V are well fitted by the Gaussian, i.e.

and at _{
G
} ≤ –16 V by the Lorentzian, i.e.

as shown in Fig. 2(c) and (d), respectively, where * e* is the elementary charge,

*is the plank constant,*h

T

_{ L }and

T

_{ R }are the tunneling probabilities at the left and right tunneling barriers,

*is the full width at half maximum,*Γ

*is the Boltzmann constant,*k

*is the temperature,*T

ε

_{ F }is the Fermi level, and

ε

_{ 0 }is the quantum level.

The shape of the Coulomb oscillation peaks must be Gaussian, which is attributed to thermal broadening, while the resonant tunneling current peaks must be Lorentzian, which is attributed to energy uncertainty (Radosavljevic et al., 2002). Therefore, the d_{
D
}/ d_{
D
} oscillation at _{
G
} ≥ –16 V in Fig. 2(a) should be Coulomb oscillation peaks, and at _{
G
} ≤ –16 V in Fig. 2(a), they should be resonant tunneling current peaks. In other words, the device operates in particle nature mode at _{
G
} ≥ –16 V, and in wave nature mode at _{
G
} ≤ –16 V.

Fig. 3 shows the d_{
D
}/d_{
D
} characteristic as a function of the drain voltage _{
D
} at 7.3 K and _{
G
} = –12.795 V. The d_{
D
}/d_{
D
} peak spacing of 26 mV is observed in the plot, which corresponds to the quantum energy level separation in the SWNT quantum island. The separation of the quantum energy levels is indicated by

where * r* is the radius of the SWNT,

*is the length of the SWNT, and*L

ν

_{ F }is the Fermi velocity (Kamimura & Matsumoto, 2004). When

*becomes large, the equation becomes*n

and shows a constant value independent of * n*.

The energy separation _{
Q
} of the quantum levels for an SWNT length of 73 nm is calculated to be 24 mV from eq. (2). Because this estimated value of energy separation of 24 mV is in good agreement with the d_{
D
}/d_{
D
} peak spacing of 26 mV in Fig. 3, it can be concluded that the entire SWNT channel acts as a single quantum island.

Fig. 4 shows a contour plot of the d_{
D
}/d_{
D
} characteristic as a function of _{
G
} and _{
D
} at 7.3 K. The characteristic can also be divided into two modes, the particle nature mode and wave nature mode. At _{
G
} ≥ –16 V, as shown in Fig. 4(a) and (b), the plot clearly shows the Coulomb diamond structures, which are getting smaller with negatively increasing _{
G
}. Additionally, line shaped quantum levels are observed outside of these Coulomb diamond structures. Therefore, at _{
G
} ≥ –16 V, the device operated in the particle nature mode. As shown in Fig. 4(b), at –16 ≥ _{
G
} ≥ –20 V, the Coulomb blockade was lifted and the Coulomb diamond structures disappeared at around _{
G
} = –16 V. The quantum levels still remain and the so-called Fabry-Perrot quantum interference pattern is observed at this region. Finally, at _{
G
} ≤ –20 V, the quantum levels are getting blurred with negatively increasing _{
G
}, as shown in Fig. 4(c). Therefore, at –16 V ≥ _{
G
}, the device operated in the wave nature mode.

Fig. 5(a) shows the Coulomb charging energy _{
C
} as a function of _{
G
}. _{
C
} is obtained by

where * ΔE* is estimated from the size of the Coulomb diamond.

E

_{ C }drastically decreases with a negatively increasing

V

_{ G }, almost reaching zero at

V

_{ G }≥ –16 V. Therefore, the Coulomb diamonds disappear at –16 V ≥

V

_{ G }. Tunneling capacitance

C

_{ t }and gate capacitance

C

_{ G }as a function of

V

_{ G }are shown in Fig. 5(b).

C

_{ t }and

C

_{ G }are obtained from

and

respectively. _{
t
} depends on the thickness of the tunneling barrier and _{
G
} depends on the gate structure. _{
t
} drastically increases with a negatively increasing _{
G
}, while _{
G
} is almost constant, independent of the changing _{
G
}. The drastic increase in _{
t
} is attributed to the decrease in the thickness of the Schottky barriers at the contacts between the SWNT quantum island and electrodes (Javey et al., 2003). When the Schottky barriers become thin with negatively increasing _{
G
}, as shown in the inset of Fig. 5(b),
_{
t
} drastically increases and the coupling strength of the wave function between the outside and the inside of the Schottky barrier becomes stronger. Therefore, the Coulomb blockade is lifted and _{
C
} becomes zero at -16 V≥ _{
G
}.

The dependence of the full-width at half maximum (FWHM) of the resonant tunneling current peak characteristic on _{
G
} is shown in Fig. 6(a).

The FWHM is estimated from an _{
D
}-_{
G
} plot using the Voigt function, which is a convolution of the Gaussian and Lorentzian functions, and can be used to divide the FWHM into the FWHM of the Gaussian _{
G
} and that of the Lorentzian _{
L
}. Fig. 6(b)–(d) shows several curves fitted by the Voigt function, where the solid line is the experimental data, the dotted lines are the fitting curves, the dashed line is the cumulative fitting curves, and each alphabetic marker for peaks A-F corresponds to a marker in the plot of Fig. 6(a).

In the particle nature mode of _{
G
}≥ –16 V, _{
L
} is negligibly small and the _{
G
} of about 0.2 V is independent of _{
G
} because the FWHM of the Coulomb oscillation mainly depends on the thermal broadening of Fermi dispersion (Foxman et al., 1993). On the other hand, in the wave nature mode of –16 V ≥ _{
G
}, _{
G
} becomes negligibly small and _{
L
} increases linearly with the negatively increasing _{
G
}. _{
L
} is proportional to the tunneling probability as follows:

where α is the ratio of the modulated energy to applied _{
G
}, and * T* is the tunneling probability (Kamimura & Matsumoto, 2006). Therefore, the increase in

w

_{ L }seen in Fig. 6 represents an increase in the tunneling probability, which is attributed to the decrease in the thickness of the Schottky barriers with a negatively increasing

V

_{ G }.

The logarithmic dependence of the drain current _{
D
} on the inverse of the temperature is shown in Fig. 7.

### 3.2. Hysteresis elimination

Fig. 8 shows the temperature dependence of differential conductance as a function of the gate voltage.

The quantum levels peaks became blurred with increasing temperature. However, even at * T* = 100 K, the quantum levels still remained, as indicated by arrows.

The static characteristic measurement of the devise with silicon dioxide layer on the SWNT channel shown in Fig. 9 was carried out using an Agilent B 1500.

In the measurement, the data were integrated for a few seconds to eliminate the effect of noise, which was set by the equipment automatically. After that, the data were recorded. In the drain current _{
D
}-gate voltage _{
G
} characteristics of the SWNT multi-functional quantum transistor under a drain voltage of 11 mV at 7.3 K, the drain current showed a periodic peak- and valley structure with two periods. As shown in Fig. 9(a),
_{
D
} was observed only in the negative _{
G
} region, which indicates that the measured SWNT was a p-type semiconductor. Moreover, a large period of ~3 V at around _{
G
}=0 V was observed, which was attributed to the Coulomb oscillation characteristic. A part of Fig. 9(a) is expanded in the horizontal scale and shown in Fig. 9(b), where small period of _{
G
}=0.65 V was observed at a higher _{
G
}, which was attributed to the Fabry-Perot interference property (Liang et al., 2001) of hole, the coherent length of which was more than 73 nm, and coherent transport in the entire channel was achieved. Fig. 9(c) shows a contour plot of the second-order differential conductance as a function of gate and drain voltages, in which a clear Fabry-Perot interference pattern can be seen.

From the small period of drain current oscillation of _{
G
}=0.65 V and the equation (Kamimura & Matsumoto, 2006) of

where _{
Q
} is the quantum energy separation, * α* is the gate modulation coefficient,

*is Plank’s constant,*h

ν

_{ F }is the Fermi velocity in graphene,

*is the length of the cavity of the hole, and*L

*is the elementary charge, we estimated the width of the quantum well and obtained the length of the cavity of the hole*e

*to be 55 nm. The estimated value is in agreement with the channel length 73 nm of the device which was obtained by scannig electron microscope (SEM) (inset of Fig. 1). The slight difference between the estimated*L

*and the channel length must be attributed to the width of band bending in the SWNT e.g., Schottky barriers.*L

In Fig. 9(d), the gray (black) line shows drain current vs. applied gate voltage from -40 (40) to 40 (-40)V in a forward (backward) sweep of gate voltage. The SWNT multi-functional quantum transistor showed almost no hysteresis characteristics. The SWNT multi-functional quantum transistor was covered with a silicon dioxide layer, which prevents the SWNT from absorbing and releasing molecules in ambient, because fluctuation of molecules on the SWNT usually induces the hysteresis characteristics in the drain current-gate voltage characteristics. Moreover, the SWNT multi-functional quantum transistor has a small channel length of 73 nm, which reduced the number of trap sites near the SWNT and, at the same time, the number of trapped carriers causing hysteresis characteristics. We believe that the purification was effective to reduce the hysteresis characteristics. Thus, the hysteresis characteristic in the drain current–gate voltage characteristic was eliminated.

### 3.2. Single charge sensitivity of SWNT multi-functional quantum transistor

Figure 10(a) shows the time dependence of the conductance of the SWNT multi-functional quantum transistor at 7.3 K with a gate bias of _{
G
}=-25.36 V. A short sampling time of 10 ms was set in the dynamic characteristic measurements shown in Fig. 10.

The applied gate voltage was under the Fabry-Perot interference region. The SWNT multi-functional quantum transistor shows RTS, as shown in Fig. 10(a), The RTS showed three levels, n, n+1 and n+2, of the conductance shown in Fig. 10(a). At a lower applied gate voltage, current levels higher than n+2 such as n+3 and n+4 appeared. The multiple levels of RTS are attributed to charge fluctuating charge storages near the conduction channels of the SWNT multi-functional quantum transistor. Moreover, because there was a single-charge storage including multiple energy levels or were some charge storages being at almost the same distances from the conductance channel of the SWNT multi-functional quantum transistor, the RTS appeared. Figures 10(b) and 3(c) show histograms of the conductance levels of RTS at _{
G
}= -25.36 V and _{
G
}= -25.39 V, respectively. The three peaks of conductance of RTS expressed as P_{n}, P_{n+1}, and P_{n+2} are shown in Figs. 10(b) and 10(c). The conductance levels of the three peaks directly correspond to the conductance levels of RTS. On the other hand, the relative heights of the peaks correspond to occupation probabilities at each conductance level of the RTS. The heights of the peaks depend on applied gate voltage. _{
n
} decreases and _{
n+1
} and _{
n+2
} increase with slightly increasing applied gate voltage from _{
G
}= -25.36 to -25.39 V, which means that the energy levels in the charge storage are modulated by applied gate voltage.

The gate voltage dependences of the natural log of the ratio between the _{
th
} peak and the (_{
th
} peak in the conductance histogram _{
m+1
}
_{
m
} (m= n, n+1, n+2, , n+4) are shown in Fig. 11(a). The natural log of _{
m+1
}
_{
m
} linearly depends on applied gate voltage and saturates in each _{
G
}. Each starting point of saturation is marked by an arrow in Fig. 11(a). The charge storage energy levels are floating. Therefore, modulations of energy by _{
G
} may be different at each charge storage. We believe that the reason why the natural log of _{
m+1
}
_{
m
} saturates at each _{
G
} may depend on the difference in energy modulation by _{
G
} at each charge storage.

The charge transition is modeled, as shown in Fig. 12(a), in which the energy barrier is between the SWNT and the charge storage. Figures 11(b)-11(e) are enlargement plots of each _{
m+1
}
_{
m
} (m= n, n+1, n+2, , n+4). The energy differences _{
n
} between the charge storage energy level _{
n
} and the Fermi level _{
f
} are expressed as _{
n
}= _{
f
} -_{
n
}
* ,* which is modulated by the applied gate voltage

V

_{ G }. According to equilibrium statistical mechanics,

P

_{ m+1 }

/P

_{ m }is given by (Peng et al., 2006)

where _{f} and g_{S} are the degeneracy of the top of valence band and the charge storage, respectively. g_{f}/ g_{S} is assumed to be 1. * β* is 1/kT.

E

_{ n }includes the contributions of e electrostatic potential induced by

V

_{ G }, intrinsic energy level in the storage, and Coulomb charging energy. The basis of eq. (1) is the Arrhenius equation. In this model, the height of the barrier is the energy difference between

E

_{ f }and

E

_{ n }. Assuming a linear dependence of

ΔE

_{ n }on

V

_{ G },

ΔE

_{ n }can be written as

ΔE

_{ n }=

*(*αe

V

_{ 0 }−

V

_{G}), where

*is the gate modulation coefficient, and is a constant value of 0.062.*α

*is obtained from the periods of Fabry-Perot interference characteristic on*α

V

_{ D }and

V

_{ G }.

V

_{0}is the offset voltage, and is obtained from the intersecting point of the extrapolating line of the fitting lines and the line of ln(

P

_{ m+1 }/

P

_{ m })= 0. Therefore, eq. (1) is transformed to

Equation (2) is the transformed Arrhenius equation, in which V_{G} is the parameter.

From the dependence of P_{n+1}/P_{n} on V_{G} shown in Fig. 11 and eqs. (1) and (2), ΔE_{n} can be obtained, and is shown in Fig. 12(b). The obtained energy levels are from 1.57 to 1.79 eV.

## 4. Conclusion

In summary, we succeeded in fabricating and demonstrating a multi-functional quantum transistor using the particle nature and wave nature of holes in SWNT. This transistor can operate in the wave nature mode as an RTT and in the particle nature mode as an SHT. We were able to reveal that the principle of the characteristic transition from an SHT to an RTT is the modulation of the coupling strength between the SWNT quantum island and the electrodes by the applied V_{G}.

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