Self-resemble ( 自我相似 ) NanoLetters, Vol. 7, No. 2, p , (2007).

R Time Electrical connections Silver electrodes SWNT film Self-resemblance

Clean room Summer SWNT film Thermometer 7 days

SWNTs Peak-to-peak and valley-to-valley Positive temp coefficient of resistivity (PTCR, metallic) Heat island ( 熱島效應, 夏天 ) 室外最熱時為 36  C at 2-4 pm 最涼為  C at 4-6 am ( 中央氣象局 )

PTCR 是可理解 因為 O 2 doping O2O2 EFEF Charge transfer Doping state, gap closing, metallic

不能理解的是為何 SWNTs 具高熱感性, 甚至可以感應溫差只有  1 度 的變異, 而銅不能 ? PTCR  α tube = per K and α Cu = per K 二者相差無幾 為何 SWNTs 之熱感性質比銅更敏銳 !

Structure Bonding W/m*K (ref1) thermal conductivity (at R.T.) sp 3 sp 2 with interlayer coupling 250 W/m*K (ref3) sp W/m*K (ref2)

sp W/m*K (ref4) W/m*K (ref4) sp 2 with interlayer coupling Structure Bonding thermal conductivity (at R.T.)

40000 W/mK K Temp (k) A single (10, 10) tube (ideal model) 100 K 400 K50 K 6000 W/mK 1. Between k, K is temp dep 2. Why a change-over emerges at 100 k? K = C v · ·l <100 k, l  constant, so K is dominated by C v. >100 k, C v  constant, so K is dominated by l, and l decreases as temp increase, due to umklapp process, Diamond: k = 2500 W/mK Graphene: k = W/mK (PRL, 84, 4613, 2000 by Tomanek)

Quantized phonon and phonon speed reaches 10,000 m/s Science, 289, 1730, 2000 by Hone et al Ph 1 Ph 2 Ph 3 EFEF C1C1 C2C2 C3C3 V1V1 V2V2 V3V3 Electron band structure EgEg Op 1 Op 2 Op 3 A1A1 A2A2 A3A3 Phonon band structure EgEg Ph 3 > Ph 2 > Ph 1

RBM D and G bands 沿碳管軸向之熱傳導主要二個 phonon modes 高頻 低頻

Phonon modes at high frequency in a CNT

Thermal contraction of CNTs Tube length Tube volume  : thermal linear expansion coefficient  : thermal volumetric expansion

以上 data 得知 a single CNT 之熱傳很快 ( 散熱也很快 ) 因此不可能出現熱感性質比銅金屬靈敏之現象 Phonon generation (ps) 快於 phonon-electron coupling (ns) ( 熱能還來不及傳給電子就已經 dissipated)

SWNTs 以 bundle form 方式出現 1. 碳管尺寸螺旋性不一致 無法形成完美六方堆積晶格 2.Radial breath mode (RBM) 于 bundle form 中變得很重要

完美六密堆積 一致性之 RBM 熱傳由 1D (single tube)  3D (bundle) Heat flux Phonon dispersion from 1 D into 3D structure and heat conduction continues along tube axis (normal process).

Ph 1 Ph 2 向量合仍然在 BZ 內 (normal process) no heat resistance and heat conduction proceeds forwards

不完美堆積 RBM 不一致

Bundle becomes heat resistance (Umklapp process) Heat flux reduction

Heat resistance Reciprical lattice

Bundle bcomes heat reservoir ( 熱儲存槽 ) 熱儲存槽可合理解釋為何碳管比銅更具熱感性 環境溫度

1.Enhanced resistive phonon 2.Heat flux reduction 3.Low thermal conductivity 4.Heat reservoir Bundle 成為熱儲存槽之三個證據 1 可由 3 來直接驗證 single carbon nanotube, k = W/mK at room temperature SWNT film = W/mK Phys. Rev. B, 59, 2514 (1999)

2. Heat flux reduction Heat flux Heat flux reduction k = .M.KQ/L M: characteristic number of occupied phonon branches (=1.5πk B TR 2 /ha ) KQ: thermal conduction quantum (=π2k B 2 T/3h = 9  W/K) L: are bundle length (=1-2 μm) h: Planck’s constant R: tube radius (= 0.7 nm) A: tube-tube separation (= nm) k = 1.3   W/K (bundle) k = 6  W/K (single SWNT) Phys. Rev. Lett, 95, (2005).

4.Heat reservoir 熱儲存槽意味碳管溫度比外界溫度 高的時後更高 低的時候更低

熱儲存槽 吸熱 放熱 吸熱 放熱 / 放熱 = 96% 高效率之熱交換器

In vacuum Negative temperature coefficient of resistivity, (NTCR, semiconductor) O 2 desorption from nanotubes O 2 doping state EFEF VBVB CBCB EgEg

See video

Single tube Bundle medium spacing Bundle large spacing Hardening effect Vibration spectra

碳管間距小

熱由左傳到右

碳管間距大

Phonon density of state single

Phonon density of state 7.31

Phonon density of state 6.71

Heat capacity at 300K Single38

P cri =  2 EI/L 2, P cri :critical axial load, E: elastic modulus, L: beam length, I: inertia moment (=A  2 )  : radius of gyration A : beam cross section

5 meV/atom 0.2 eV/atom

1. 彎曲 bundle 變直 2.Thermal vibration (8-10 s -1, phonon wave through rigid body, 剛硬 ) 3.Bundle lengthening by 13.5%

E1E1 E2E2 Tube displacement + Tube Rolling Applied electric field intensity = tube-tube friction

Cohesive energy of unfolded tube (planar sheet) E 1 = 3-5 meV/atom E 2 = eV/atom