PHYSICAL REVIEW LETTERS

PRL 98, 166803 (2007)

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Terahertz Current Oscillations in Single-Walled Zigzag Carbon Nanotubes Akin Akturk,* Neil Goldsman, and Gary Pennington Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA

Alma Wickenden Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783, USA (Received 26 September 2006; published 19 April 2007) We report time-dependent terahertz current oscillations on an n
10 single-walled zigzag carbon nanotube (CNT) that is 100 nm long. To obtain transport characteristics in this CNT, we developed an ensemble Monte Carlo (MC) simulator, which self-consistently calculates the electron transport and electrical potential. The ensemble MC simulations indicate that, under certain dc bias and doping conditions, the average electron velocity and concentration oscillate. This leads to current oscillations in space and time, on the tube, and at the contacts. We attribute this to accumulation and depletion of the CNT electrons at different locations on the tube, giving rise to low and high density electron regions. These local dipoles are a result of intra- and intersubband scatterings and different subband dispersion relations. This in turn forms propagating dipoles and current oscillations. DOI: 10.1103/PhysRevLett.98.166803

PACS numbers: 73.63.Fg, 73.22.f, 78.67.Ch

I. Introduction.—Carbon nanotube (CNT) electrical behavior has been theoretically and experimentally investigated [1–13]. Research indicates high electron drift velocities and large electron mobilities as compared with silicon. In addition to high drift velocities, earlier steadystate Monte Carlo (MC) simulations predict spatially dependent velocity oscillations under the presence of uniform applied electric fields. The periods of these oscillations are calculated to be in the range of tens of nanometers in single-walled zigzag 100 nm semiconducting tubes [4]. The presence of these velocity oscillations may lead to THz oscillators that are important for application in future wireless communication electronics. However, it must first be determined if these spatial oscillations will translate into time-dependent current oscillations in CNTs that are under dc bias. To investigate whether these time-dependent current oscillations do indeed occur, we built upon previous work [1– 4] and developed an ensemble MC simulator for CNTs. The MC simulator solves the semiclassical transport equations and the Poisson equation self-consistently inside the CNT. The ensemble MC calculations provide the potential and electron current, velocity, and concentration profiles as functions of space and time on the tube for an applied dc bias. Our calculations show the existence of terahertz current oscillations on the tube under several bias and tube doping conditions. We attribute these time- and space-dependent oscillations to intra- and intersubband electron-phonon couplings, influences from subband dispersion curves, and the spatial-temporal variation in the potential. II. CNT energy spectra.—To investigate self-sustained oscillations in CNTs, we developed an ensemble MC simulator and used it to examine a 100 nm long singlewalled zigzag semiconducting carbon nanotube with fundamental indices of n; m
10; 0 and a diameter (d) of approximately 0.8 nm. To initialize the MC calculations, 0031-9007=07=98(16)=166803(4)

we input the CNT subbands structure. The CNT subbands are obtained by applying zone-folding methods to the graphene energy spectra, which was calculated using the tight-binding model [9]: s















































































      Tk 

cos  4cos2 : Ek;

 1  4 cos 2 n n (1) Here k,
, and T are the wave vector along the tube’s axis, the wave vector index around the tube’s circumference, and the length of the zigzag tube’s translational ˚ ), respectively. We take the value of vector (4.31 A nearest-neighbor -hopping integral  as 3 eV [1,2]. Additionally, we here account for the lowest three CNT subbands in two valleys [1,2]. We next incorporate the longitudinal acoustic and optical phonon dispersion curves that are obtained using the fourth nearest-neighbor tight-binding model and zonefolding methods [2]:           Ep q; 
Ep0   @vs  jqj   (2)  :    d  Above q is the phonon wave vector,  is the azimuthal quantum number [14], Ep0  is the energy of phonon associated with  at zero momentum, vs is the sound velocity in graphene,  is a step function, and  is zero for optical phonons and one for acoustic phonons. The radial breathing mode (RBM) [15,16] is also incorporated into calculations. Continuum modeling is used to determine the energy spectrum and deformation potential of the RBM [17]. For an n; m
10; 0 single-walled CNT, the RBM energy and deformation potential per
length have been calculated to be 35 meV and 2:3 eV=A, respectively, which are in relatively close agreement with other calculated values [15]. Furthermore, we do not con-

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© 2007 The American Physical Society

PRL 98, 166803 (2007)

PHYSICAL REVIEW LETTERS

sider Kohn anomalies for nondegenerately doped semiconducting CNTs. These anomalies can be important for the metallic tubes [18]. III. CNT scattering.—From the electron and phonon spectra, we calculate electron-phonon scattering rates using the Fermi’s Golden Rule and the deformation potential approximation. Here we include inter- and intravalley, acoustic and optical, emissions and absorptions for the lowest three subbands. The electron-phonon scattering rates are very high due to the large densities of states, especially at subband minima. The doping and electron concentrations on the tube are very low resulting in small electron-electron and impurity scattering rates and are neglected. The resulting electron-phonon scattering rate from k in subband
k~
k;
 to k0 in subband
0 k~0
k0 ;
0  via a phonon with momentum qq
k0  k in subband 

0 
 2n [1,2,4] is written below:  2 2  ~ k~0 
@D Q NEp q;   1  1 
Ek0 ;
0  k; 2 2 LEp q;   Ek;
  Ep q; :

(3)

Here D is the deformation potential taken to be 9 eV, obtained by comparison to experiment [1,5,6]; Q is 2 a for q

























2  2 2 optical phonons and q   a n  for acoustic phonons
L is the tube’s length;  is the linear mass (a
2:49 A); density; and N is the Bose-Einstein phonon occupation number at equilibrium. ~ k ~ is calculated by The total scattering rate from k, 0 ~ taking the integral of Eq. (3) over k or equivalently k0 and
0 , using the following density of states:       Ek0 ;
0    : gEk0 ;
0 
 (4)  0  0  
2 T sinTk  cos    2 n For example, the resulting total intrasubband scattering rate from k~ to k~0 via an acoustic phonon branch with a dispersion relation of Ep q
jqj@vs (
0) is  X D2 jqj  1 1 1 ~ k
  jqj@vs 2 2 q 2vs exp E   1 th    Ek;
 jqj@v      s   ; (5)     Tkq 
2 T sin   cos  2 n where Eth is the thermal energy at room temperature. Solutions of the energy conservation formula for phonon momenta q, written below, provide phonons that are allowed in the above summation. Ek  q;
  Ek;
  jqj@vs
0:

(6)

For all other phonon momenta on this phonon branch, the corresponding scattering rates are zero due to the delta function given in Eq. (3). IV. Monte Carlo calculations. —In the ensemble Monte Carlo simulations, electrons are injected into the

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CNT from both ends. The injected electrons, as well as the electrons that are initially in the tube, are sampled from a Fermi-Dirac distribution. Electron dynamics then follow semiclassical transport, with the state variables updated according to equations of motion every t seconds, where t is small compared with both the mean free time between collisions and the dielectric relaxation time. At every t, we solve the Poisson equation as well, to update the internal electric field distribution. This process continues until the relevant physical phenomena emerge. A. Time independent results.—We first compared lowfield MC simulation results obtained for very long intrinsic CNTs with published experimental [5–7,10,11] and theoretical [1,2,12,13,19,20] data: reasonable agreement was obtained. For example, we calculated low-field mobilities for CNTs, with approximate diameters of 1 nm, 2 nm, and 3 nm, as 6 103 , 1:5 104 , and 3:6 104 cm2 =V s, respectively. In Ref. [10], investigators measured 3 103 , 5 103 , and 1:1 104 cm2 =V s, for CNTs ranging in diameter from 1 nm to 3 nm, with the chirality of the tube unspecified. In Ref. [6], low-field mobility of a 3.9 nm diameter CNT was experimentally determined to be 5–7 104 cm2 =V s, and Ref. [7] measured 2:5 104 cm2 =V s for the electron mobility of a 2 –2.4 nm diameter CNT. We also calculated electron velocities which are similar to those given by [13]. We next simulated a 100 nm long n
10 CNT with a constant field equal to 90 kV=cm. We observed positiondependent velocity oscillations due to intrasubband electron-phonon scatterings [4]. Specifically, when electrons gained roughly 160 meV and 200 meV energies above the energy band minima, they tended to scatter back to the subband minima after emitting one of these phonons. The electron scatterings were associated with large densities of states at the subband minima of these quasi-one-dimensional systems. This was creating the space-dependent oscillations [4]. Similar oscillations have also been predicted by [19,20]. B. Time- and space-dependent results: current oscillations.—To investigate the space and time dependence of the electron transport, we used a nondegenerately N-doped CNT, such that the Fermi level was at 70% of the half band gap above the midgap. (Here, doping is used to increase the coupling between the electrostatic potential and the net charge density.) Initially both ends of the tube were at ground level. We then ramped the voltage on the right side from 0 to 0.9 V linearly in 1 ps and later fixed this side at 0.9 V. Calculated space and time dependencies of CNT currents are shown in Fig. 1, with the range of current values given by the color bar on the right. P We here calculate current at location z using Iz
e k;
ne k;
; zvk;
, where e is the electronic charge, ne is the linear electron density associated with momentum k and subband
in the vicinity of z, and v is the electron velocity in subband
for momentum k: vk;

1@ @Ek;
 @k .

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FIG. 1 (color). Calculated CNT electron currents in amperes as a function of time and position on the tube. The color bar on the right indicates the range of the current amplitude. Frequency domain representations calculated using the Fourier transform of time signals between 1 ps and 9 ps, of points 1 nm apart on the tube, are also plotted on the right. The main oscillation frequency of the current is 17.2 THz.

FIG. 2 (color). Calculated CNT electron concentration profiles as a function of time and tube location, in the subbands with the lowest energy minimum (subband 1, on the left) and second lower energy minimum (subband 2, on the right). Most of the first subband electrons scatter to the second subband between 20 nm and 30 nm, and then scatter back to the first subband after 40 nm. The color bars below the plots show the ranges of electron concentrations.

Numerical results in Fig. 1 indicate current oscillations with spatial periods of approximately 30 nm and temporal periods of roughly 0.1 ps. To investigate the time signals at different points on the tube with the current profile shown in Fig. 1, their frequency domain representations are obtained using a discrete Fourier transform. These curves for the time signals from 1 ps to 9 ps sampled every 4 fs, of points that are 1 nm apart on the tube, are shown on the right of Fig. 1. There is a pronounced oscillation at 17.2 THz. There are also weaker oscillations at the harmonics: 2 and 3 times the main frequency. Furthermore, the amplitude of the noise level is approximately 40 times smaller than that of the main frequency, indicating clear oscillatory characteristics (signal/noise 40). Here the amplitude of the highest oscillations is 60 nA, superimposed on a dc level of roughly 40 nA. Self-consistent solutions with the Poisson equation indicate the formation of dipole charge distributions. These dipoles are due to intrasubband scatterings, intersubband scatterings between the first and the second subbands, and the influence of the subband dispersion curves. The concentration of electrons in the first and the second subbands, as a result of intersubband scatterings, is compared in Fig. 2. Specifically, this figure shows that most of the electrons are initially in the first subband as they enter the tube from the left, or the grounded side. Because of the applied potential induced electric field on the tube, most of the electrons gain energies high enough to scatter from the first subband to the second subband after they travel about 20 nm to 30 nm, as can be seen in Fig. 2. When electrons have sufficient energies, MC calculations indicate that they tend to scatter from the first subband to the energy minimum of the second subband, where the density of states

peaks. (Here the electron concentration in the third subband is very low, and therefore its contribution to the total current is negligible.) Also, around the subband minima, electrons have low velocities, increasing charge separation. The electrons in the second subband later start scattering back to the first subband in significant numbers about 40 nm away from the grounded side. Finally most of them scatter back to the first subband as they approach the right end of the tube, where the positive bias is applied. Thus the relative low and high density of electrons at different locations, due to fast and slow moving electrons, create relative charge dipoles. Each relative dipole is superimposed on a larger applied electric field and thus propagates along the tube giving rise to current oscillations. The electron-phonon scatterings within a subband as well as the influence of the subband dispersion curves also contribute to the formation of relative dipoles and thus current oscillations. To investigate whether current oscillations will sustain themselves in the absence of intersubband scatterings, the system is also simulated only considering the intrasubband scatterings. Calculated CNT electron currents as a function of time and tube location are shown in Fig. 3(a), with the current range given on the right. CNT current oscillations are again observed. Using a discrete Fourier transform, frequency domain representations of time signals from 1 ps to 9 ps sampled every 4 fs, of points that are 1 nm apart on the tube, are calculated. They are plotted in Fig. 3(a) employing the same scale for magnitude as that of Fig. 1. The signal to noise ratio of these oscillations is approximately 37. To examine what happens in the absence of electronphonon scatterings, we simulated the tube again under the same conditions, except now all scatterings are disabled.

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also occur when only intrasubband scatterings and the lowest subband are considered. The signal to noise ratio when three subbands are included is slightly higher than when only one subband is accounted for. In contrast, the simulation of transport with no scattering shows very limited resonance. Simulations suggest that the source of most of the oscillations can be attributed to resonant scattering, while band structure curvature and multivalley effects also contribute. The presence of intrinsic THz oscillations in CNTs may open new opportunities for the introduction of sources into this underutilized band of frequencies. We are grateful to Professor Michael Fuhrer at the University of Maryland for useful discussions.

FIG. 3 (color). Calculated CNT electron currents in amperes as a function of time and position on the tube, considering (a) only the intrasubband scatterings and (b) no scatterings. The color bars on the right indicate the ranges of the current amplitudes of the plots on their lefts. Frequency domain representations calculated using the Fourier transform of time signals between 1 ps and 9 ps, of points 1 nm apart on the tube, are also plotted on their rights. For the no scattering case, there are no pronounced current oscillations.

The resulting calculated CNT electron currents as a function of time and space are shown in Fig. 3(b), with the current range given on the right. Here there are current variations due to the statistical nature of the MC calculations, the ringing in local potentials as a result of external bias application, and the nonlinear increase in electron velocity as a function of energy. However, these variations in current values do not give rise to consistent current oscillations throughout the tube. Frequency domain representations of time signals are shown on the right of Fig. 3(b). Here there is a weak oscillation, but the noise level or the statistical fluctuations dominate the current characteristics (signal/noise 5). V. Conclusion.—We investigated electron transport in a 100 nm long single-walled zigzag CNT under dc bias. Investigations indicate that throughout the tube there are scattering enabled self-sustained current oscillations at terahertz frequencies. Oscillations occur when three subbands and intra- and intersubband scatterings are all included in the calculations. In this case, electron transfer between the lowest two subbands is predicted. Oscillations

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