Phonon wave-packet scattering and energy dissipation dynamics in carbon nanotubeoscillatorsMatukumilli V. D. Prasad and Baidurya Bhattacharya Citation: Journal of Applied Physics 118, 244906 (2015); doi: 10.1063/1.4939277 View online: http://dx.doi.org/10.1063/1.4939277 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/118/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The performance volatility of carbon nanotube-based devices: Impact of ambient oxygen Appl. Phys. Lett. 95, 123118 (2009); 10.1063/1.3236779 Deformation potential carrier-phonon scattering in semiconducting carbon nanotube transistors Appl. Phys. Lett. 90, 062110 (2007); 10.1063/1.2437127 Characterizing energy dissipation in single-walled carbon nanotube polycarbonate composites Appl. Phys. Lett. 87, 063102 (2005); 10.1063/1.2007867 Double‐Resonant Raman Scattering in an Individual Carbon Nanotube AIP Conf. Proc. 685, 225 (2003); 10.1063/1.1628023 Neutron scattering studies of carbon nanotubes AIP Conf. Proc. 486, 299 (1999); 10.1063/1.59801

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Phonon wave-packet scattering and energy dissipation dynamics in carbonnanotube oscillators

Matukumilli V. D. Prasad1 and Baidurya Bhattacharya2,a)

1Advanced Technology Development Centre, Indian Institute of Technology Kharagpur, Kharagpur,West Bengal 721302, India2Civil Engineering Department, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721302,India

(Received 15 July 2015; accepted 19 December 2015; published online 31 December 2015)

Friction in carbon nanotube (CNT) oscillators can be explained in terms of the interplay between

low frequency mechanical motions and high frequency vibrational modes of the sliding surfaces.

We analyze single mode phonon wave packet dynamics of CNT based mechanical oscillators, with

cores either stationary or sliding with moderate velocities, and study how various individual

phonons travel through the outer CNT, interact with the inner nanostructure, and undergo

scattering. Two acoustic modes (longitudinal and transverse) and one optical mode (flexural

optical) are found to be responsible for the major portion of friction in these oscillators: the

transmission functions display a significant dip in the rather narrow frequency range of 5–15 meV.

We also find that the profile of the dip is characteristic of the inner core. In contrast, radial

breathing and twisting modes, which are dominant in thermal transport, display ideal transmission

at all frequencies. We also observe polarization dependent scattering and find that the scattering

dynamics comprises of an oscillating decay of localized energy inside the inner CNT. This work

provides a way towards engineering CNT linear oscillators with better tribological properties.VC 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4939277]

I. INTRODUCTION

Friction during the relative motion of two atomic surfa-

ces arises from an exchange of energy between the surfaces.

For strongly interacting surfaces, friction involves large

stress and displacement fields, rearrangement of interatomic

bonds, and loss or exchange of atoms at the interface. In case

of weakly interacting interfaces, which is the focus of this

work, friction amounts to the dissipation of energy through

phonon excitations1–5 and electronic excitations.6,7

Cumings and Zettl’s experimental work8 on multi-

walled carbon nanotube (MWCNT) pull-out set the stage

for CNT based oscillators, actuators, and bearings. While

telescoping out the inner walls, they quantified the dynamic

(static) friction force between the walls as less than

4.3� 10�15 N/A2 (less than 6.6� 10�15 N/A2), which are

almost three orders smaller than friction forces in current

MEMS.9 Around the same time, Kolmogorov and Crespi’s10

computational work on smooth intershell mobility between

the walls of commensurate MWCNTs introduced the possi-

bility of NEMS operating at Gigahertz range.11 In a recent

experimental work,12 this ultra-smooth sliding under ambient

conditions is observed even in a centimeter long DWCNT.

And very recently, Ma et al.13 proposed an analytical model

for the critical length above which such superlubricity

breaks down. Various nanoscale mechanical devices have

been proposed since then that take advantage of the relative

motion of carbon nanotube walls, such as: nanobear-

ing,10,14–17 nanoactuator,18 nanoswitch,19 nanomotor,20,21

nanogear,22 nanobolt-nut pair,23,24 nanoaccelerometer,25

nanoresonator,26 nanothermometer,27 oscillator,8,28,29 vari-

able resistor,30 high performance FETs,31 Brownian motor,32

inertial measurement systems,33 non-volatile memory devi-

ces,34–36 nanoscale cargo transport,37 etc.

During the first half of the previous decade, various stud-

ies based on classical molecular dynamics have attempted

to address the role of commensuration,10,38–40 boundary condi-

tions,41 functionalization,42 defects,43–46 interwall dis-

tance,45,47 length of sliding surfaces/overlap/telescoping,45,48

and temperature38,49–52 in energy dissipation in CNT based

linear oscillators. There have been conflicting findings in

regard to sliding forces in commensurate vs. incommensurate

systems. Incommensurate systems were found less dissipative

than commensurate systems;10,38,39 however, Tangney et al.40

claimed that dynamical friction was independent of commen-

suration of bitubes and of the overlap area between tubes.

They further claimed that the source of friction was the motion

of tube ends (surface edges), which depended on the relative

velocity of tubes. Zhao and Cummings53 investigated these

differences and found the outcome to depend on temperature,

commensuration, and type of molecular dynamics ensemble

adopted. In an attempt to resolve the conflicting findings on

the effect of commensurability, Zhu et al.54 identified interac-

tion strength and intertube energy corrugation as two crucial

parameters in assessing the effect of commensurability on

energy dissipation, yet the underlying physics need to be

investigated.

Stone-Wales and vacancies have been found to excite

low frequency vibrational modes45 and to markedly increase

the dissipation rate.44,46 However, a single Stone-Wales

a)Author to whom correspondence should be addressed. Electronic mail:

baidurya@iitkgp.ac.in

0021-8979/2015/118(24)/244906/9/$30.00 VC 2015 AIP Publishing LLC118, 244906-1

JOURNAL OF APPLIED PHYSICS 118, 244906 (2015)

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defect was found to suppress dissipation in a linear oscillator

by avoiding rotational motion.43 Although dissipation rate

was typically found to increase linearly with temperature in

most studies, Chen et al.52 reported an inverse relation at

ultra-low temperature ranges (�0.01 K–1 K): the critical

temperature at which the inversion occurred corresponded to

the saturation of thermal lubrication and the appearance of

phononic friction. The critical temperature in turn was

related inversely with the CNT length.

Phononic friction at the interfaces involving adsorbed

monolayers has been studied both experimentally and theo-

retically. The experimental works involved hydrogen and

deuterium terminated Si surfaces,55 H2O, N, superheated He

films adsorbed on Pb(111) substrates,56 and adsorbed films

of several inert gas atoms and hydrocarbons on various me-

tallic substrates as reported in Ref. 57. The theoretical works

involved CO molecule adsorbed on the (100) face of Cu.58

Although a few pioneering experimental works8,12,28,59 have

addressed friction in sliding nanotubes, these are all phenom-

enological in nature, and to our knowledge, phononic friction

in CNT oscillators has not yet been subjected to theoretical

or experimental investigations.

Recent theoretical attempts60,61 on thermal energy

exchange at the interfaces of CNT confirmed a strong de-

pendence of frequency and polarization of phonons and

also of interface characteristics. In the area of interfacial

thermal transport, a variety of CNT interface configura-

tions such as surface-surface interface,60,62,63 end-end

interface64 and CNT-substrate interface61 have been stud-

ied. In order to understand the effect of interface bonding

and spacing on thermal transport, most of these studies an-

alyzed individual phonon scattering mechanism at the

interfaces. In an MD based study, Greaney and

Grossman63 observed a resonance based energy transfer

between CNTs when positioned side by side. Kumar and

Murthy60 used heat pulse propagation and wavelet analysis

to identify dominant phonon modes in thermal transport

across CNTs when CNTs were positioned perpendicular to

each other.

In CNT oscillators, deformations such as bending and

wavy modes of outer CNT, which are manifested by the

collective excitation of disorderly phonon modes, have

been identified as the major channels through which the

translational kinetic energy of the core dissipates as ther-

mal energy.48,50,65 Exchange of vibrational energy from

useful mechanical motion to thermal fluctuations and

vice-versa have so far not involved any phonon level

treatment. Phonon wave packet formalism66 provides a

powerful computational framework for treating single pho-

non transport across interfaces.64,66,67 Using this scheme,

the dynamics of single mode phonon at the sliding CNT

interface can be studied.

Once the motion of core initiates, a typical oscillation

cycle involves: motion along the outer CNT where a part

of orderly translational energy dissipated in to disorderly

high frequency phonon modes; reversal of motion of core

at the ends of outer CNT. The focus of the present work

is on the energy exchange dynamics between orderly

translation motion of core and phonon modes of outer

CNT. The fundamental understanding of the interaction

mechanism between individual phonon modes and orderly

moving core is of crucial importance in nanoscale

tribology.

In this work, we present a phonon wave-packet simula-

tion based study of the scattering mechanism of individual

phonons in pristine CNTs, in order to elucidate their role in

phononic dissipation occurring in co-axially sliding oscilla-

tors. Although, due to sliding instabilities, all phonon modes

are involved to some degree in the dynamics of dissipation,

here we study only those modes individually that are known

to be dominant in thermal transport. High transmittance indi-

cates less dissipation at the encounter of a single phonon and

core.

II. COMPUTATIONAL DETAILS

Phonons exist over a wide range of frequencies. Low

frequency phonons show a plane wave type behavior because

of their long wavelength (as in the propagation of sound);

whereas phonons of high frequency show a particle type

behavior (as in the case of light-matter interaction). Phonon

modes and other physical properties such as specific heat,

dielectric,68 piezoelectric,69 elastic constants,70,71 etc., can

be explained in terms of harmonic travelling waves in the

atomic lattice (anharmonic forces would be required for

obtaining properties such as thermal conductivity, phase

transitions, etc.).

Phonon wave packet method is used in this work along

with molecular dynamics simulations to study the influence

of phonon scattering on the dynamics of the CNT linear os-

cillator. Phonon wave packet method has been used by

Schelling et al.66 to study scattering mechanism at the coher-

ent semiconductor interface, by Refs. 64, 66, and 67 to study

the influence of phonon scattering due to defects67 and func-

tionalization64 on the thermal transport along CNTs, and by

Helgee and Isacsson72 to investigate the phonon transmission

across the grain boundaries in graphene. However, to our

knowledge, phonon wave packet method has not been

applied to study energy exchange mechanism in CNT

oscillators.

The force constant matrix is obtained using the supercellmethod64 in which the atoms in the unit cell are displaced

one at a time (small enough displacement to maintain har-

monicity) and resultant forces on all atoms of the supercell

are computed. Fourier transform of this force-constant ma-

trix gives the dynamical matrix, diagonalizing which results

in phonon frequencies and phonon modes at a particular

wave vector. The relation between frequencies and wave

vector gives the dispersion relation.

Phonon wave packets are formed from the linear combi-

nation of vibrational eigenstates. To generate wavepacket

with wavevector q0 and centered at z0, the initial displace-

ment (u) and initial velocity (t) of the kth base atom in lthunit cell along the direction a, assuming a Gaussian spread,

are set as

uslka ¼

Affiffiffiffiffiffimkp

Xq

exp �q� q0ð Þ2

2r2

� �es

ka qð Þexp iq zl � z0ð Þ� �

;

244906-2 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)

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tslka ¼

Affiffiffiffiffiffimkp

Xq

�i2pxs qð Þexp �q� q0ð Þ2

2r2

� �

� eska qð Þexp iq zl � z0ð Þ

� �;

where A is the wave amplitude, r is the broadening parame-

ter to control width of Gaussian phonon wave packet (we

choose r to be 2p/100a where a is size of unit cell, such that

the wave packet covers a width of 200 unitcells), zl is posi-

tion of the lth unitcell, and mk is the mass of the kth atom.

xsðqÞ, eskaðqÞ are the frequency and eigenvector correspond-

ing to polarization s and the wave vector q.

The propagation of the wave packet thus generated on

the outer CNT with its corresponding group velocity is simu-

lated. Molecular dynamics simulations are conducted using

LAMMPS.73 As in previous works on phonon wave packet

simulations of CNTs,64,74,75 the polymer consistent force

field (PCFF)76,77 is adopted to model the atomic interactions.

As the parameters in this potential are developed using abinitio computations corresponding to 0 K, PCFF is a better

choice for MD simulations involving ground state systems.

All simulations are conducted in NVE ensemble with a time

step of 1 fs. To ensure that anharmonic effects are com-

pletely avoided, the ground state (�0 K) configuration is

adopted in all simulations. Scattering due to the inner nano-

structure is visualized. Energy transmission coefficient is

computed as the ratio of transmitted energy to the incident

phonon wave packet energy. We simulated the scattering

process until a stable kinetic energy is attained on both sides

of the interface region to ensure that the process of interac-

tion is completed within the simulation time. Thus, the simu-

lation time span of a molecular dynamics run varied from

30 ps for high group velocity phonons to 300 ps for low

group velocity phonons. Wave packet amplitude of 0.5 A is

used throughout all the simulations. (For a detailed sensitiv-

ity study to arrive at this 0.5 A amplitude, see Fig. S1.78)

Fig. 1 shows the phonon dispersion plot for a (10,10)

SWCNT unit cell (2.45 A long and consisting of 40 atoms),

calculated using the supercell method. Of the 120 possible

phonon modes, only four are acoustic in nature, and being re-

sponsible for almost half of the thermal energy transport,79

are studied here in detail: the longitudinal acoustic mode

(LA), one of doubly degenerate transverse acoustic mode

(TA) and the twisting mode (TW), which is unique to nano-

tubes due to their symmetry. The remaining 116 are the opti-

cal modes; of these, two are picked up for further study on

account of having the highest group velocities: the radial

breathing mode (BR) which extends to most part of the fre-

quency spectrum than any other mode, and the first order

flexural optical mode (FO). A micron-long version of the

same unit cell forms the outer CNT of the oscillator used in

this study, as discussed next.

III. RESULTS AND DISCUSSION

The oscillator consists of an open ended 50 nm long

(5,5) SWCNT core within a micron long (10,10) SWCNT

outer tube. Fig. 2 shows the initial locations of the wave

packet and the inner core. Along with moving cores with

sliding velocities of 100 m/s and 300 m/s, we also studied

stationary cores. We restrict the core velocity to 300 m/s in

order to avoid resonance effects. Corresponding to each of

the five phonon modes selected for study (LA, TA, TW, BR,

and FO), a phonon wave packet is generated in the outer

tube far away from the inner core (500 unit cells to the left

of the centre) and we study how the individual phonon trav-

els through the outer CNT, interacts with the inner nano-

structure, and undergoes scattering. A typical phonon

wave packet interaction process with inner core is depicted

in Fig. 2 using wave packet displacement components corre-

sponding to before, during, and after the interaction with

inner core.

A. The enablers of ideal transmission

Fig. 3 shows the transmission coefficients of the five

phonon modes for the oscillator system as a function of the

FIG. 1. Phonon dispersion and symme-

try of phonon modes in (10,10) CNT

calculated from the PCFF force con-

stants using supercell method. The

highlighted five dispersion curves (LA

in orange, TA in cyan, TW in red, BR

in green, and FO in pink) are studied in

this work. v is frequency, “a” is uni-

tcell size, and h is Planck’s constant.

Symmetry of phonon modes (LA, TA,

TW, BR, and FO modes in right

panel).

244906-3 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)

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phonon energy. The group velocity (the slope of the disper-

sion plot in Fig. 1) is plotted along the axis on the right.

Clearly, the TW and BR modes show near ideal transmission

throughout the entire frequency range. In fact, the radial

breathing mode is the most characteristic vibrational mode

for CNTs and can help identify the chirality and diameter of

the CNT through Raman scattering experiments. In this

mode, all atoms vibrate radially in-phase and resemble the

breathing effect.

Like the radial breathing mode, the twisting mode too

shows near ideal transmission (0.95 and above) in almost the

entire range of frequencies (Fig. 3(c)). With minimal radial

displacements, atoms in this mode vibrate tangential to the

surface of the CNT. Importantly, group velocity80 and pho-

non transmission (even in the presence of a vacancy

defect67) in the low energy region of TW mode have been

found to be unaffected by the diameter of the outer CNT,

which is unlike in other acoustic phonon modes. We conjec-

ture that such diameter-independence extends to phonon

transmission in the presence of inner core as well, provided

the inner core maintains a stable interwall distance. BR and

TW modes show ideal transmission irrespective of the slid-

ing velocity of the core.

The TW and BR modes are the in-phase circumferential

and in-phase radial/breathing displacements respectively as

shown in Fig. 1. We believe that this symmetry in displace-

ments, coupled with the rotational symmetry in the defect-

free core, is responsible for the complete absence of scatter-

ing in the TW and BR modes. In contrast, a significant

amount of scattering has been observed in the presence of

mismatched symmetries where these TW and BR modes

interact with defects81 or functionalizations75 instead of pris-

tine inner CNT.

The three other modes—two acoustic (LA and TA) and

one optical (FO)—are responsible for major part of the fric-

tion in CNT oscillators. These modes are studied in detail.

FIG. 2. Top: Snapshots of a propagating LA wave packet along the outer CNT

at times 1 ps, 11 ps, and 20 ps. Transverse and axial displacements are drawn in

horizontal (blue) and vertical (red) planes respectively. The position of the

inner CNT is represented by the two dotted lines at the center. Bottom: The

atomic configuration of coaxially sliding CNTs with imposed wave packet on

left side of the moving core. Outer tube is shown in cut-out section view.

FIG. 3. Phonon transmission functions

for (10,10) CNT with (5,5) CNT-L50

as stationary and moving core.

Transmission coefficients (left vertical

scale) and phonon group velocity (blue

lines) (right vertical scale) as a func-

tion of incident phonon energy, hv. (a)

Longitudinal acoustic phonons. (b)

Transverse acoustic phonons. (c)

Twisting acoustic phonons. (d) Radial

breathing optical phonons. (e) First

order flexural optical phonons.

244906-4 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)

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B. The dissipative modes

In case of CNTs with surface defects,64,67,75,81 the

acoustic phonon modes are known to transmit ideally near

the C-point (i.e., at low wave numbers) irrespective of their

group velocity, which is exactly what we see in Figs. 3(a)

and 3(b) of this work. A similar behavior is also seen in pho-

non transmission at Si/Ge interface.82 As the frequency

increases, however, there appears to be no consensus regard-

ing the extent of correlation between transmission coefficient

and group velocity in acoustic modes (Refs. 67, 75, and 81

as opposed to Refs. 64 and 74). Our study (Fig. 3(a)) shows

negligible correlation between either nLA or nTA and vg in

low energy regions, but a significant positive correlation

beyond the point of lowest group velocity.

The acoustic phonons with wavelengths very close to

C-point are long enough to not interact with the inner nano-

structure. Moving just right of the C-point, in the low energy

range (5–15 meV), they are able to interact with the core and

thereby lead to significant scattering (down to �0.7 for LA).

nTA is even smaller than nLA in this region, and can come

down to 0.2. As one moves up the energy scale, nLA and nTA

feature a sharper dip that is centered around the energy cor-

responding to the least group velocity. Finally, an ideal trans-

mission region is realized in the high energy region

(23–35 meV). The core velocity is not influencing the trans-

mission in low energy region. In the high energy region, the

transmission functions are deviating from the stationary

case. In Fig. 3(a), the small dip at 25 meV in nLA is amplified

significantly for both sliding velocities. Moreover, for

300 m/s case, the amplified dip is shifted towards the low fre-

quency. In contrast, the dip in nTA at 18 meV (Fig. 3(b))

found evaded for moving core cases. The changes in the

high frequency range transmission functions of moving cores

indicate the strong interaction of high frequency phonon

modes with the moving core.

Bending and wavy modes has been identified as the

major source of energy dissipation in CNT oscillators.48,50,65

In general, TA mode vibrations would manifest as bending

of CNT by transverse deformation of the tube structure.

Although the role of TA phonon modes has been studied for

CNT resonators,83 thermalization of CNTs,84 and thermal

energy transport in CNTs with defects67,81 and functionaliza-

tions,75 their role in energy exchange in CNT oscillators has

not be studied until now.

The optical phonon modes with low group velocity has

been predicted earlier60 as strongly interacting modes

between two CNTs. The optical phonon modes with wave-

lengths in the intermediate regime and having large enough

group velocity are found responsible for non-Fourier heat

conduction85 in SWCNTs. In Fig. 3, the optical modes, how-

ever, behave differently near the C-point compared to the

acoustic modes. The general characteristic of optical pho-

nons—low transmission near C-point—is displayed by both

FO and BR modes (Figs. 3(d) and 3(e)). The strong scatter-

ing of optical modes near the C-point is due to their low

group velocities causing them to be confined in the defective

or edge region of the inner CNT. Therefore, the optical pho-

nons of long wavelength cause substantial heat generation

through electron-phonon interactions. Similar to LA and TA,

nFO also show a sharp dip in low energy region at 11 meV.

The next dip at 15 meV is found shifted towards the high fre-

quency side due to the core motion (the 100 m/s case). For

higher core velocity, this dip is less pronounced. The small dip

in nBR at 110 meV corresponds to the band anticrossing region

(where these BR modes mix strongly with other bands).

For dissipative modes (LA, TA, and FO), the transmis-

sion functions in the low energy region are almost unaffected

by the core velocity. In the moderate energy region, trans-

mission functions are changed significantly due to strong

interaction of phonons with the moving core (nLA in

22–28 meV, nTA and nFO in 13–22 meV). Similar trends have

been observed in earlier studies86,87 at finite temperatures.

We conjecture this coupling at moderate frequencies occurs

due to resonance. Further, in the high energy region, the

transmission functions are unaffected and all three dissipa-

tive modes transmit ideally irrespective of core velocity.

The increased scattering of slower LA phonons is well

agreeing with earlier studies.75,81 The interface scenario in

the CNT oscillator system provides a new perspective to

comprehend this behavior. The severity of scattering in low

group velocity region for LA phonons in our oscillator can

be explained by considering both phase velocity vp and

group velocity vg as presented in Fig. 4. However, in an ear-

lier study by Wang and Wang,81 the scattering of slower LA

phonons in CNTs with Stone-Wales defects (no inner core)

has been investigated. Using an infinite atomic chain as simpli-

fied model, they demonstrated the correlation between low

transmission and low group velocity as a general characteristic

of phonon transport across CNT with broken translational

invariance. Their classical acoustics based reasoning is that a

low group velocity in the LA mode causes the phonon wave

impedance to be low, which in turn causes slower phonons to

almost completely be reflected by the defect.

From the Fig. 4, we conjecture that a high value of the

ratio vp/vg is the reason for the huge scattering at low group

velocities. In the LA mode, atoms vibrate mainly along ra-

dial and axial directions. For the duration of time when the

packet passes over the inner tube, packets with the slower

group velocity have a higher number of radial oscillations.

FIG. 4. Ratio of phase velocity to group velocity. vp/vg (red) on right y axis;

and on left y axis, phase velocity (black) and group velocity (blue) of LA

mode as a function of incident phonon energy, hv.

244906-5 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)

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This number of oscillations over the inner tube is directly

correlated with (vp/vg). Fig. 4 clearly shows a sharp increase

of vp/vg corresponding to the least group velocity.

From the results of transmission functions of five impor-

tant phonon modes, we can observe ideally transmitting BR

and TW modes and a narrow frequency band (5–15 meV) in

which the major scattering occurs for three dissipative

modes. The TA and FO modes show a severe scattering in

this frequency band. The sliding velocity significantly modi-

fying the transmission functions of all three dissipative

modes, however not in the peak scattering region.

C. Dynamics of the scattering events

Further, to elucidate the frictional dissipation mecha-

nism, we now look at the scattering events created as a

phonon wave packet travels along the outer tube of the oscil-

lator and encounters the inner core. In each case, the excita-

tion starts at 500 unit cells left of the center of the stationary

core and travels right. Fig. 5 shows the time evolution of ki-

netic energy along the length of the oscillator (taken to be

the kinetic energy of the first basis atom of each unit cell).

The top row corresponds to the LA phonons, the second row

to the TA phonons, and the last row to the FO phonons. In

each of these nine panels, the square plot corresponds to

energy time history of the outer CNT while the companion

strip corresponds to the core. It should be noted that the color

scales corresponding to the nine panels are all different.

Phonons with three different energies are studied for

each mode. For the LA mode, we choose three phonons

(5.76, 9.18, and 12.3 meV) that outline the profile of the first

dip in the transmission plot occurring to the right of the

FIG. 5. Phonon scattering dynamics. Temporal evolution of the phonon wave packets while interacting with (5,5) CNT-L50 as inner core. LA phonons with

energy (a) 5.76 meV, (b) 9.18 meV, and (c) 12.3 meV. Decay of localized phonon state (Quasi bound state) in inner CNT can be clearly seen. TA phonons with

energy (d) 2.4 meV, (e) 8.15 meV, and (f) 17.76 meV. Both extremes of scatterings can be seen from (d) and (e). FO phonons with energy (g) 8.78 meV, (h)

11.3 meV, and (i) 13.54 meV. The kinetic energy per atom (meV) is represented according to the color bar. Contour maps for both outer and stationary inner

CNTs are shown separately. Vertical dashed lines indicate the position of inner CNT, contour map of which presented separately on right side.

244906-6 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)

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gamma point in Fig. 3(a). For the TA mode, we choose one

close to the gamma point (2.4 meV), one at the first dip

(8.15 meV), and one near the lowest vg (17.76 meV) in the

transmission plot (Fig. 3(b)). For the FO mode, we again

choose three phonons (8.78 meV, 11.3 meV, and 13.54 meV)

that outline the profile of the first dip in the transmission plot

(Fig. 3(c)). Although these three modes are responsible for

energy dissipation in CNT oscillators, each mode has its own

characteristic behaviour distinctly different from the other

two.

In the LA mode, the wave packet creates the first scatter-

ing event in the outer tube only when it travels over the right

end of the core. A substantial part of the packet is reflected

back first through the inner CNT, part of which then is sent

back along the outer CNT as it reaches the left end of the

core. This creates a cycle of slowly decaying scattering

events and represents anomalous reflection of LA phonon by

the inner core.

Although the two dips in the transmission function are

very similar to those in the LA mode, the scattering events in

the TA mode are significantly different from those in LA.

The packet starts at 500 unit cells to the left of the centre

as before. While the long wavelength TA phonon shows no

scattering (Fig. 5(d)) as it passes over the core, the one near

low vg shows diffused scattering (Fig. 5(f)). For the TA pho-

non with energy near the first dip (middle panel), scattering

occurs as soon as the packet reaches the left end of the core,

whereas for the LA phonon with the same energy profile,

scattering occurs only when the packet has travelled the

length of the core.

The first order optical phonons too behave very differ-

ently from the LA mode. Not only is there a substantial

reflection as soon as the phonon encounters the inner core,

but also when the phonon passes over the right end of the

core. The diffused scattering however in the right panel is

similar to that observed for TA phonons with the lowest

group velocity.

Similar to the TW and BR modes, the LA mode also has

rotational symmetry in match with the symmetry of inner

CNT. Surprisingly, the LA mode is not showing ideal trans-

mission. From the scattering events (Figs. 5(a)–5(c)), first we

can observe the unscattered passage of the LA mode phonon

wave packet over the left end atoms of inner CNT which is

similar to the TW and BR modes. Afterwards, the dominated

longitudinal component of the LA mode wave packet causes

an intense shearing that leads to transmission of wave packet

energy into inner CNT. The transmitted part of wave packet

continues to propagate in the inner CNT and causes an end

scattering due to right end atoms. The reflected part of wave

packet propagates towards left end while part of its energy

transmitting to the outer CNT. This oscillating decay of the

LA mode wave packet in the core, which can be clearly seen

in the contour strips corresponding to the inner core.

For the TA and FO modes, such a matching symmetry

does not exist. Once the propagating phonon encounters the

mismatched symmetry or augmented degrees of freedom due

to the presence of inner CNT (at the ends itself), it generates

quasibound vibrational states in the inner CNT. The gener-

ated low quasibound state acts as localized scatterer and

causes mixing of modes and severe scattering. From the

trends of TA and FO modes, we can predict the dissipation

due to other unstudied optical modes is in proportion with

their transverse atomic displacements. The BR and TW

modes are found to display near ideal transmission irrespec-

tive of the sliding velocity which indicates that the phonon

vibrational modes with matched symmetry with core (unlike

dissipative modes) causes no scattering. Hence, such phonon

modes can be recognized as the enablers of ultra-low dissipa-

tion in coaxial sliding of CNTs.

IV. CONCLUSIONS

We have studied single mode phonon scattering dynam-

ics in co-axially sliding CNTs. The radial breathing and

twisting modes (which are relatively high group velocity

phonons) are potentially the major contributors to ultra-low

friction in CNT oscillators. Over a wide range of frequencies

and for core velocities up to moderate values (�300 m/s),

they show near ideal transmission. Extensive testing with

wavepacket amplitudes in the feasible range (0.1–1 A) con-

firmed that the BR mode’s ideal transmission is not affected

by wavepacket amplitude.

In contrast, the major participants in phonon dissipation

in CNT oscillators are two acoustic (longitudinal and trans-

verse) and one optical (first order) modes. We found the

energy ranges of phonons that can prevent ideally smooth

sliding of CNTs. The transmission plot in the LA mode fea-

tures two dips, and the first one (at low phonon energies, in

the 5–15 meV range) is characteristic of the core. The maxi-

mum scattering observed in the TA mode (�0.9) is higher

than that in the LA mode (�0.5), which suggests the

dominance of out of plane deformations in the damping of

oscillations. An anomalous reflection of LA phonon is

observed. Up to moderate velocities of the moving core (up

to 300 m/s), the essential scattering behavior in the LA, TA,

and FO modes is not changed; however, an amplification or

suppression of few high frequency modes is observed due to

the moving core.

We restricted the core velocity to a moderate value of

300 m/s in this work since at very high core velocities

(between 800 and 1100 m/s), we found the BR mode of the

outer CNT to become excited (not shown in this paper) simi-

lar to other works.87 We conjecture that such high amplitude

resonance in the BR mode engenders LA phonons, which in

turn causes a high frictional dissipation.

Although the present simulations are conducted on a

ground state configuration, the deformations imposed by the

thermal effects, wavy and sagging deformations of long

CNTs (which essentially create the out of plane deformation

on the walls of outer CNT) causes further scattering. Thus,

the dissipation behavior could be affected by the deforma-

tions due to structural instabilities. Since the anharmonic

effects indulge in observing single mode wave packet, study-

ing finite temperature scattering behavior of single mode

phonons is not possible. However, the present trends, in

essence, corroborating the previous MD simulations based

results.40

244906-7 M. V. D. Prasad and B. Bhattacharya J. Appl. Phys. 118, 244906 (2015)

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In this study, we restricted the core geometry to an

uncapped (5,5) CNT of 50 nm long. Our future work will

address the effect of core geometry in detail by considering

different core structures. Preliminary results suggest that a

certain amount of core geometry dependence may exist on

phonon scattering dynamics. For example, when the length

of open ended (5,5) CNT is doubled the first dip in the LA

transmission mode appears to shift towards the C-point and

the corresponding phonon scattering increases in magnitude.

Also, when the armchair CNT is replaced by a zigzag CNT

of almost the same diameter, the first dip in LA transmission

appears to become more pronounced indicating an increased

scattering by the zigzag core. Preliminary results also appear

to suggest the exciting possibility that the transmission plot

can be used to identify the inner core structure using its

unique signature.

Our findings may lead to better design of sliding surfa-

ces and to make highly efficient functional devices. Being

able to drive or cool specific phonon modes88,89 in order to

attain high phonon mean free path or to have controlled

anharmonic phonon dynamics is a promising way to engi-

neer the performance of CNT coaxial oscillators. However,

controlling individual phonons in situ remains extremely

challenging.

ACKNOWLEDGMENTS

We thank Dr. Vikas Varshney and Dr. Jonghoon Lee for

insightful discussions. Some of the MD runs were performed

on PARAM YUVA HPC for which we thank Centre for

Development of Advanced Computing (CDAC), Pune.

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