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Graphene conductivity A lot of effort has been devoted to the question of transport in pure graphene due to the remarkable fact that the dc conductivity is finite without any dissipation process present. Recall the Drude theory: 0 J (t ) ne 2 1 ne 2 ( ) , where 0 E (t ) m 1 i 1 i m 1 4 ne 2 p (in simple metals, 0 ) 4 m 4 4 ne 2 2 p where p plasma frequency. m In Graphene the density of states at the Fermi level vanishes, so n should be small. However, is not well known but surely very large. So we can see that the problem of evaluating 0 is not trivial. 2 M. Lewkowicz and B. Rosenstein, PRL 102, 106802 (2009) Dynamics of Particle-Hole Pair Creation in Graphene find: 4 e2 lim lim 1 disorder 0 0 h e2 1.27 h e2 e2 1.57 h lim lim 2 0 disorder 0 2 h They support this value of the dc conductivity of pure graphene Other authors find e2 3 h There is no accepted theoretical value Measurement of conductivity They report =1.7 1 Previous experiments on Si substrates give 4 1 There is no accepted experimental value We study the time-dependent transport through Graphene using J mn (t ) Tr[ fJˆmn (t )] f 1 ( H0 ) e 1 Jˆmn (t ) in Heisenberg picture (M.Cini 1980) , 4 semiinfinite semiinfinite a 2.46 Angstroms After sudden switching of voltage we calculate the current 1 J mn (t ) Tr[ fJˆmn (t )] f ( H0 ) , Jˆmn (t ) in Heisenberg picture. e 1 J mn (t ) Conductivity (t)= V e2 0 h d 2 at t t * L vF J |E 0 E d 1 I |V 0 V 6 first plateau = e2 W 2h L 4e 2 second plateau = independent of geometry h Fullerenes Sir Harold W. Kroto, University od Sussex,Nobel Prize for Chemistry in 1996 Discovery September 4,1985 Was known initially as soccerene Fullerenes consist of 20 hexagonal and 12 pentagonal rings as the basis of an icosohedral symmetry closed cage structure. 20 hexagons 12 pentagons F 32 20 hexagons 12 pentagons 20*6 12*5 180 2S S 90 g 0 Euler : F S V 2 V 60 8 In theory, an infinite number of fullerenes can exist, their structure based on pentagonal and hexagonal rings, constructed according to rules for making icosahedra. Il fullerene non è molto reattivo data la stabilità dei legami simili a quelli della grafite ed è inoltre ragionevolmenteinsolubile nella maggioranza dei solventi. I ricercatori hanno potuto aumentare la reattività fissando dei gruppi attivi alla superficie del fullerene. 9 per produrre i fullereni: arco elettrico, a circa 5300°K, con una corrente elevata e bassa tensione, utilizzando elettrodi in grafite in atmosfera inerte (argon) a bassa pressione. 10 Endohedral compounds They are fullerene cages with La or other metal atoms inside. Some have been crystallized and found to superconduct 11 The art of hitting the goal with every shot We have observed de Broglie wave interference of the buckminsterfullerene C60 with a wavelength of about 3 pm through diffraction at a SiNx absorption grating with 100 nm period. This molecule is the by far most complex object revealing wave behaviour so far. The buckyball is the most stable fullerene with a mass of 720 atomic units, composed of 60 tightly bound carbon atoms. http://www.univie.ac.at/qfp/research/matterwave/c60/index.html 12 Carbon Nanotubes Fascinating electronic and mechanical Properties: 1. Depending on their chiralities, nanotubes can be metallic, semimetallic or semiconducting 2. Remarkably high Young’s moduli and tensile strength “Imagine the possibilities: materials with ten times the strength of steel and only a small fraction of the weight!” ------Former resident Bill Clinton 13 14 Multi-Walled NanoTube (MWNT) 15 S. Iijima. "Helical microtubules of graphitic carbon." Nature 354 56 (1991) 16 Carbon Nanotubes S. Iijima, Nature 354, 56 (1991) 17 Carbon Nanotubes: Lattice Structure From Wikipedia 18 Carbon Nanotubes: Lattice Structure S. Iijima, Nature 354, 56 (1991) L≈1m d≈nm Graphene sheet Nanotube Single Wall ( m, n) CNT with a , b primitive lattice vectors, na mb roll-up-vector of ( n, m) CNT ( atom at xa yb is identified to the one at ( x n)a ( y m)b ) Depending on n,m the CNT can be metal or semiconductor 19 This is a possible choice of the basis which is often used: a1 b | a1 || a2 | a 3 a2 (m, n) translation in Graphene Rmn ma1 na2 Rmn identified with R00 (m, n) nanotube. Then, (n,0) nanotubes are called zigzag nanotubes, and (n,n) nanotubes are called armchair nanotubes. Otherwise, they are called chiral. 20 axis of CNT a1 (n,0) alias Zigzag CNT a2 path towards the tip (4,0) zigzag CNT (15,0) zigzag CNT path around the belt: 2n atoms 21 armchair CNT a1 a2 CNT axis = y axis (2, 2) armchair CNT armchair CNT path along the y axis All armchair nanotubes are metallic, as suggested by paths along axis 22 “Armchair” geometry (n,m) with m=n, always metallic “Zig-zag” geometry (n,m) with m=0 e.g. (5,0),(6,4),(9,1) are semiconducting • “Chiral” geometry •all the rest 23 The alternative basis which we used for the band structure of Graphene is also in use for CNT a1 a2 3 3 3 3 a1 a( , ), a2 a( , ), a1 a2 3a 2 2 2 2 3 3 In this basis, (n, m) translation Rnm na1 ma2 a( (n m), (n m)) 2 2 (n, m) CNT : Rnm identified with R00 . Since both conventions are used we must be ready to handle both of them. Zigzag CNT path around the belt: 2n atoms 24 a1 Zigzag CNT using alternative basis 3 3 3 3 a1 a( , ), a2 a( , ), a1 a2 3a 2 2 2 2 a2 (4, 0) zigzag CNT becomes (4, 4) in new basis CNT axis = x axis a 2 3 a a 3 The circumference is na 3 Indeed, the shift along the belt of CNT is obtained with m n, Rn, n a(0, n 3) . The wavevector k around belt is quantized: k na 3 2 , na 3 The CNT radius 2 integer 25 pz Electronic bands of (n,-n) zigzag CNT-tight-binding approximation Since the wavevector k k y around belt is quantized: k na 3 2 , integer, just insert into Graphene band structure 2 E (k ) E0 J 0 ky k 3 3 3 k y a . k y a 4 cos 1 4 cos k x a cos 2 2 2 2 , 0, 1, 2, na 3 3 k y a cos cos and one finds, setting k x q (along axis) n 2 band structure: E (q, ) E0 J 0 3 cos 4 1 4 cos qa cos n n 2 2 26 Carbon Nanotubes as quasi 1D systems: one component of k quantized • Band Structure of graphene • NT: Compact transverse dimension Discretization of k Subbands correspond to different values of k k|| is a continuous variable 27 k|| 27 Note: 3 band structure: E (q, ) E0 J 0 1 4 cos qa cos 4 cos 2 n n integer quantum number, (n,n)=chirality index 2 3 dependence along x: cos qa since the distance along x between two A atoms 2 3 3 is a. Setting c= a 2 2 1d BZ q c c 3 with c a , 2 Note also: E (q, ) E (q, ) 28 E (q, ) E0 J 0 1 4 cos qc cos 4 cos n n 3 BZ q , c a. Since cos cos E (q, ) E (q, ) c c 2 n n 2 Note: the (4,-4) zigzag CNT has 8 atoms around the belt. Generally, (n,-n) zigzag 2n atoms in belt 2n bands Examples: (n,-n) zigzag: for n 5, 10 bands: 0, 1, 2 each occurs twice because of double sign in E ; no more bands since 3 yields cos( 3 2 ) cos( ). 5 5 for n 6, 0, 1, 2,3 each with E 12 bands , 0,3 nondegenerate since cos( 3)=0. 6 29 3 1 4 cos qc cos 4 cos , c a 2 n n 2 The dispersion E (q, ) E0 J 0 3 implies two nondegenerate bands E ( q, 0) E0 J 0 5 4 cos qa , 2 n n and for even n, , two flat bands E ( q, ) E0 J 0 2 2 Zigzag CNT are Metals for n multiple of 3. 1 cos( ) . Therefore : 3 2 1 cos if n 3 and 1, if n 6 and 2,.... if n 3* | | ; then, n 2 2 radicand 1 4 cos qc cos 4 cos 2(1 cos qc ). n n qc 1 cos( ) Using cos , E E0 2 J 0 | cos( ) | 2 2 2 at the border of the BZ, qc , E E0 for both bands. No gap! 2 30 From Mahan’s nutshell book : band structure of a (5,0) zigzag nanotube. Labels indicate angular momentum values 31 From Mahan’s nutshell book : band structure of a (6,0) zigzag nanotube. Labels indicate angular momentum values. If m-n is a multiple of 3 the nanotube is metallic. 32 armchair CNT a1 a2 CNT axis = y axis (2,2) armchair CNT armchair CNT path along the axis All armchair nanotubes are metallic, as suggested by paths along axis 33 Recall Primitive vectors 1 1 cos( ) 3 2 Cartesian components are : a2 a1a2 angle i 3 a1 a2 2 3 2 3 3 3 3 3 a1 a( , ), a2 a ( , ), 2 2 2 2 too; indeed, j 3 2 3 2 k 3 3 0 k *3 | a1 a2 || a1 || a2 | | a1 || a2 | sin( ) 2 2 3 0 34 Armchairs are (n,n) using basis a1 a2 CNT axis Nanotubes: the site at = y axis 3 3 Rmn ma1 na2 a ( (n m), (n m)) 2 2 is identified R00 orthogonal vector CNT axis Tmn ma1 na2 ((n m), 3(n m)) taking m=n, armchair. Rn ,n a (3n, 0), (2,2) armchair CNT Length of belt:6a k kx (belt of CNT along x axis), k is quantized k 3an 2 q k y along y axis = CNT axis 35 Graphene : E (k ) E0 J 0 3 3 3 1 4 cos k x a cos k y a 4 cos k y a 2 2 2 for armchair CNT: set k y q, k x k 2 2 , 0, 1, 2, 3na 3 3 3 cos k y a cos qa cos(qc), c a 2 2 2 3 3 2 cos k x a cos a cos 2 2 3na n E (q, ) E0 J 0 2 1 4 cos cos qc 4 cos qc n For 0, E (q, 0) E0 J 0 [1 2 cos qc ] cross at qc 2 2 , E ( , 0) E0 3 3 armchair CNT are always metals. 36 From Mahan’s nutshell book : band structure of a (5,5) armchair nanotube. Labels indicate angular momentum values. All armchair nanotubes are metallic. 37