Conductance of carbon nanotubes functionalized with gold clusters during CO adsorption
We investigate the time-dependent electronic transport in single-walled carbon nanotubes (SWCNTs) functionalized with Au clusters on CO gas exposure. Using a tight-binding Hamiltonian and the nonequilibrium Green’s function (NEGF) formalism the time-dependent zeroth and first order contributions to the current are calculated. The zeroth order contribution is identified as the time-dependent Landauer formula in terms of the slow time variable, whereas the first order contribution is found to be small. The Green’s function for the SWCNT is derived using the equation of motion and Dyson equation technique. The conductance is explicitly evaluated by considering a form for the hopping integral which accounts for the effect of dopants on the charge distribution of the carbon atoms and the nearest-neighbor distance. The effect of dopants is also studied in terms of fluctuations by calculating the autocorrelation function for the experimental and theoretical data. These calculations allow direct comparison with the experiment and demonstrate how the presence of dopants modifies the conductance of the SWCNT measured experimentally and provide the only study of fluctuations in the sensor response in terms of the autocorrelation function.
pacs:73.23.-b, 73.63.-b, 72.10.Bg, 73.63.Fg
In the past years, carbon nanotubes (CNTs) , especially single-walled carbon nanotubes (SWCNTs) have been actively studied as a chemical (gas) sensor because of their unique nanostructure and electronic properties . Sensors make significant impact in everyday life with applications ranging from health to environment. Detection of hazardous gases using miniaturized sensing devices with high sensitivity is an open challenge in chemical sensing applications for environmental safety, including air-pollutants monitoring. As a result, research emphasis is on developing novel sensing materials and technologies. The chemical sensing capabilities of CNTs based sensors can be amplified by their functionalization. In recent years, carbon nanotubes decorated with metal nanoparticles (NPs) have attracted a tremendous amount of interest and research activity in the applications of CNTs as gas sensors. Previous studies have shown CNTs functionalized with metal NPs exhibit unique sensitivity toward various gases [3, 4, 5, 6, 7, 8, 9, 10]. The sensing capability of metal NP-decorated CNTs is based on the changes in their electrical properties induced by gas molecules adsorbed on the NP surface [3, 4, 5, 6, 7, 8, 9, 10]. The exceptional structural and electronic properties of CNTs make them a potentially ideal material for the exploration of electronic transport phenomena in low dimensional systems. In this context, nonequilibrium Greens function (NEGF) formalism provides a powerful technique for the development of quantitative models for quantum transport in such systems. The NEGF formalism was developed by Kandanoff and Baym , and Keldysh [12, 13] which provides a microscopic theory to model quantum transport in mesoscopic semiconductor systems [14, 15, 16, 17]. Electronic transport in these systems is divided into stationary and time-dependent phenomena. The results of stationary transport using the NEGF formalism have been reported by many authors [18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In these studies, a key result was a Landauer type formula [28, 29] which gives a relationship between the conductance of a mesoscopic sample, connected to the contacts by two leads, and its transmission properties. This NEGF technique was also extended to study time-dependent transport in mesoscopic systems [30, 31, 32, 33, 34, 35]. A general formulation for time-dependent transport through mesoscopic structures was introduced using time-dependent voltages in Refs. [36, 37, 38]. Hernandez et al.  presented their study of time-dependent electronic transport through a quantum dot using the NEGF formalism, whereas Kienle et al.  studied the time-dependent quantum transport through a ballistic CNT transistor in the presence of a time harmonic signal. The work presented in this paper is motivated by these time-dependent results [36, 37, 38, 39, 40] but distinguishes from these studies in terms of time-dependence which arises due to interaction of CO gas molecules with functionalized SWCNT not because of any externally applied time varying potentials.
We present a theory which explains the experimental result of the electronic transport in SWCNTs decorated with Au clusters (Au-SWCNTs) in the presence of CO gas molecules . To realize the application of SWCNTs as a gas sensor, a conceptual and quantitative understanding of the underlying mechanism in experiments is necessary. There has not been much effort to model such devices and this manuscript represents an important contribution to the field. The quantitative understanding of the experiment involves building up a theoretical model of the Au functionalized SWCNT to study electronic transport, and the development of a connection between the theoretical predictions and experimental observations. The experiment is performed for many Au clusters (having different dimensions) with 30 minutes of exposure to 2500 ppm CO gas at room temperature and a bias voltage of 0.5 V . To model the electronic transport through the Au-SWCNT system on CO adsorption, the calculations are performed for a defective (14,0) SWCNT decorated with an cluster. The model considered in the calculations is based on the tight-binding Hamiltonian which describes electrons in SWCNT. Analytical calculations of time-dependent electronic transport for the Au-SWCNT system are performed using the NEGF formalism. In the system, there are two time scales: a slow time scale () for CO gas flow and a fast time scale () for electron transport inside the SWCNT. Hence, an adiabatic expansion with respect to slow time variable and the Fourier transform with respect to fast time variable are made in the calculations, leading to the zeroth and first order contributions to the current. The zeroth order current is identified as the Landauer formula in terms of the slow time variable and the first order contribution is found to be small. We derive an explicit formula for the transmission function and calculate the conductance in terms of the time-dependent hopping integral and on-site energy. The formula is then used to compare the theoretical results with the experiment (i.e., conductance versus time plot) , by choosing a form for the hopping integral which accounts for the effect of the Au and CO on the transport properties of the SWCNT. The effect of the Au and CO molecules on electronic transport in SWCNT is also studied in terms of fluctuations by calculating the autocorrelation function (ACF) for the theoretical and experimental data.
2 Tight-binding model and the nonequilibrium Green’s function formalism
The adsorption configuration of nine CO molecules attached to the corner (i) and edge (ii, iii) sites of an cluster on a (14,0) SWCNT with a missing carbon atom defect  is shown in Fig. 1, where the C, Au and O atoms are shown in green, yellow and red. This configuration demonstrates that the Au cluster affects only the first nearest-neighbor carbon atoms of the SWCNT that causes changes in conductance.
To model the electronic transport in the system, a tight-binding model of the -SWCNT system (C) sandwiched between two semi-infinite (14,0) left (L) and right (R) nanotube electrodes is presented. The chosen model system is same as considered in the theoretical study of the experiment . The model is shown in Fig. 2 where Au atoms are indicated by yellow solid circles, C atoms are denoted by blue solid circles with an outline, and O atoms are indicated by red empty circles. The model consists of only four nearest-neighbor carbon atoms , , and of which only the the first nearest-neighbor carbon atoms and are affected by the cluster positioned at the missing carbon atom site . The second nearest-neighbor carbon atoms and remain unaffected. If more carbon atoms that are next nearest-neighbors to the cluster are considered in the model then there will be no significant change in the result as only the first nearest-neighbor carbon atoms contribute to the changes in the conductance. As a result, the effective length of the nanotube involved in transport for sensing is a few nanometers.
The operational principle of the model is based on the changes in conductance when the -SWCNT system is exposed to CO gas for 30 minutes. On exposure to CO gas, a CO molecule interacts with the cluster (positioned at site ) at an instant of time which affects the hopping of electrons from the orbital of one carbon atom to the neighboring carbon atom with the corresponding hopping integral (the hopping integral of pristine SWCNT). Then, the electron hops from to with the hopping integral , from to with , and from to with , Fig. 2. In a similar way, the hopping integral changes when other CO molecules interact with the cluster at the next instants of time and so on. This is how the time-dependence arises here without the application of a time-dependent bias. The time-dependent hopping integral leads to time-dependent Hamiltonian and Green’s functions for the SWCNT. Thus, to study time-dependent transport through such a nano-hybrid model the time-dependent NEGF formalism is well suited.
2.1 Model Hamiltonian
To calculate the time-dependent conductance in the model system we use the nonequilibrium Green’s function (NEGF) formalism. The time-dependent electronic transport through mesoscopic and nano scale systems has been addressed in the literature [36, 37, 38, 39, 40] where the time-dependence arises due to application of external time-dependent bias between the source and drain contacts, and the coupling between the leads and the central region can be controlled by time-dependent gate voltage. But this manuscript addresses the problem in which time-dependence arises due to interaction of CO gas molecules with the Au-decorated SWCNT for a given gas exposure time. Therefore, the total Hamiltonian of the model system corresponding to a semiconducting SWCNT (C) and the two left and right nanotube electrodes L and R is expressed as
where is the Hamiltonian of the SWCNT and is given as
where creates (annihilates) an electron in state m of the carbon lattice and is the on-site energy of the carbon atom, whereas is the nearest-neighbor hopping energy between the carbon atoms as a function of time. The effect of the Au and CO gas is included in the form of time-dependent on-site energy and hopping integrals because as the time changes, the number of CO molecules and hence their interaction with Au cluster changes, which results in a change in on-site energy and hopping integral leading to a change in sensor response.
The contact Hamiltonian is written as
where and are the creation and annihilation operators for electrons with momentum in either the L or R contact. It should be noted that there is no time dependence in , as in the experiment no external time-dependent bias is applied between the L and R contacts, and electrons in the contacts are non-interacting.
The coupling between the contacts and the central SWCNT in the absence of a time-dependent gate voltage is described by the tunneling Hamiltonian
In the matrix form, the total Hamiltonian is expressed as
where are the left and right SWCNT contact Hamiltonian and is the time-dependent tight-binding Hamiltonian for the central SWCNT. The matrices and are the tunneling (coupling) Hamiltonian between the two contacts and the central SWCNT.
Since the cluster at site affects only the first nearest-neighbor carbon atoms and with the corresponding hopping integral and , as a function of time, the second nearest-neighbor carbon atoms and remain unchanged with the hopping integrals . Therefore, the Hamiltonian of the Au-decorated SWCNT system with CO adsorption is of the form
2.2 Expression for the current and equation of motion
where, the lesser Green’s function is given as
Equation of motion
An expression for the can be derived from the equation of motion for the contact time-ordered Green’s function which is defined as 
where T is the time-ordering operator defined as = . The equation of motion for is expressed as
with the time-ordered Green’s function of the central carbon nanotube defined as 
which satisfies the equation of motion
Equation (12) can be further written as
The time-ordered Green’s function, Eq(10), is further written as
where is the contact Green’s functions operator for the uncoupled system .
Equation (14) gives rise to the lesser Green’s function 
where the retarded and lesser Green’s functions of the central carbon nanotube are defined as
Putting Eq. (15) in Eq. (7) and converting the sum over the momentum states in the contacts into an integral over energies, Eq. (7) is expressed as
where is the time-independent coupling between the left contact and the central carbon nanotube given as
with the density of states.
Splitting the limit over the time integral the expression for the current becomes
In matrix form, the current is expressed as
where denotes the Green’s function of the carbon nanotube in the presence of the coupling with the contacts.
Hence, the current flowing from the left/right contact to the central carbon nanotube is written as
2.3 Adiabatic approximation
In the adiabatic approximation, the time scale over which the system parameters change is large compared to the life time of an electron in the system (CNT) . In the problem of electronic transport through the SWCNT-based sensor the experimental time scale of the gas interaction with the Au-SWCNT is much longer () than the time scale of electron transport inside the SWCNT (). Therefore the adiabatic approximation is applied to study the electronic transport in the gas sensor.
Adiabatic expansion for the Green’s functions
The best way to separate slow and fast time scales is to re-parameterize the Green’s functions. In other words the time variables of the Green’s functions are replaced by a fast time difference and a slow mean time  as
The adiabatic approximation is applied to lowest order by expanding the Green’s functions about the slow time variable up to linear order in the fast time variables 
which can be further written as
where and are the zeroth and first order Green’s functions which lead to the zeroth and first order contributions to the current.
Applying adiabatic expansion on in Eq. (21) the current becomes
Consider only the zeroth order contribution to the current
Solving Eq. (26) by taking the Fourier transform and using Eq. (16) we obtain
2.4 Green’s function for the nanotube
In Eq. (13), substituting one gets
is the time-independent self-energy .
After replacing with Eq. (28) can be further written as
Equation (30) can be rewritten in full matrix form as
Applying adiabatic expansion on one obtains
Expanding around  gives rise to
Substituting Eq. (33) in Eq. (32) to get
Consider only the zeroth order contribution gives
Taking the Fourier transform with respect to the fast time variable we get
This leads to the zeroth order Green’s function for the nanotube in the presence of the coupling with the contacts
2.5 Dyson equation
Equation (30) in full matrix form is expressed as
Define two auxillary time-ordered Green’s functions and that satisfy the equation of motions
which is rearranged to
Using Eq. (39), Eq. (41) can be further written as
From Eqs. (38) and (40)
which is the Dyson equation for the system of gas sensor.
3 Results and Discussions
3.1 Calculation of the zeroth order time-dependent Green’s function
Expanding the Hamiltonian in Eq. (6) around using Eq. (33), and taking only the zeroth order contribution, the zeroth order Hamiltonian with respect to the slow time variable, is obtained. Using Eq. (37) and the Hamiltonian the zeroth order time-dependent retarded Green’s function for the SWCNT is explicitly written in a matrix form as
3.2 Zeroth order time-dependent Landauer formula
Using Eq. (27) and writing the total current as , and assuming the left and right coupling functions are proportional to each other , where is the arbitrary parameter with the constant of proportionality , a simple expression for the total current through the SWCNT is derived as
Applying the adiabatic expansion and taking the Fourier transform of the Dyson equation (Eq. (43)) for the retarded Green’s function, one gets
where the self-energy can be expressed as 
In this model system, the effect of CO interaction with the Au-SWCNT is incorporated in not in the self-energy which is time-independent and includes only the tunneling contributions with no interaction. The retarded and advanced self-energies are defined as
where and are the real and imaginary parts of the self-energy with and .
This leads to the self-energy difference as 
Using Eq. (48) the zeroth order retarded and advanced Green’s functions for the SWCNT from Eq. (37) are given as
Equation (50) gives rise to
From Eqs. (51) and (52)
Using Eq. (49), Eq. (53) becomes
Substituting Eq. (54) in Eq. (45)
In this equation, the quantity is just the self-energy (Eq. (49)). Hence the current becomes
which leads to the current
This is the time-dependent Landauer formula which is derived in terms of the slow time variable , where the transmission function of the system is identified as
with are the Fermi distribution functions in the left and right electrodes, and describe the contact-SWCNT coupling.
If only the first element of matrix and the last element of matrix are considered, the transmission function depends only on the off diagonal elements of the matrix and is expressed as
To calculate the conductance explicitly, an expression for the transmission f