Carbon Nanotubes Nanotubes for Data for Data Carbon Processing Processing Reza M. Rad Reza M. Rad UMBC UMBC Based on pages 473- Based on pages 473 -497 of 497 of “ “Nanoelectronics Nanoelectronics and and Information Technology” ”, Rainer , Rainer Waser Waser Information Technology
Introduction Introduction � Carbon Carbon nanotubes nanotubes � (CNTs CNTs) discovered by ) discovered by ( Iijma (NEC Labs), 1991 (NEC Labs), 1991 Iijma � CNT can be thought of CNT can be thought of � as a stripe cut from a as a stripe cut from a single graphite plane single graphite plane (Graphene Graphene) and rolled ) and rolled ( up to a hollow up to a hollow seamless cylinder (fig1) seamless cylinder (fig1)
Introduction Introduction � C atoms form a hexagonal network, C atoms form a hexagonal network, � 2 hybridization because of their sp 2 hybridization because of their sp 3 are mixed in, � Small contributions of sp Small contributions of sp 3 are mixed in, � due to the curvature of the network in case due to the curvature of the network in case of CNTs CNTs of � CNT diameters between 1 and 10 nm and CNT diameters between 1 and 10 nm and � micrometers long have been fabricated micrometers long have been fabricated
Introduction Introduction � CNT ends may be open or capped with CNT ends may be open or capped with � half a fullerene molecule half a fullerene molecule � Two main categories are Single Wall Two main categories are Single Wall � Nanotubes ( (SWNTs SWNTs) and Multi Wall ) and Multi Wall Nanotubes Nanotubes ( (MWNTs MWNTs) (fig2) ) (fig2) Nanotubes
Introduction Introduction � Ropes of Ropes of CNTs CNTs are frequently encountered are frequently encountered � which are self- -assembled bundles of assembled bundles of which are self SWNTs (fig 3) (fig 3) SWNTs � The small size of The small size of CNTs CNTs and their transport and their transport � properties are very attractive for future properties are very attractive for future electronic applications electronic applications
Electronic Properties Electronic Properties � Geometrical structure Geometrical structure � � The structure of The structure of CNTs CNTs is described by the is described by the � circumference or chiral chiral vector, C vector, C h , defined by: circumference or h , defined by: C h =na1+ma2 C h =na1+ma2 � Where a1 and a2 are unit vectors in the Where a1 and a2 are unit vectors in the � hexagonal lattice (see fig 1) hexagonal lattice (see fig 1) � C C h also defines P h , periodicity of the tube h also defines P h , periodicity of the tube � parallel to the tube axis parallel to the tube axis � It also settles the It also settles the chiral chiral angle which is the angle which is the � angle between C h and a1 angle between C h and a1
Electronic Properties Electronic Properties � m=n=0 : m=n=0 : chiral chiral angle is zero; tube is called angle is zero; tube is called zig zig- - � zag zag � m=n : m=n : chiral chiral angle is 30; tube is called arm angle is 30; tube is called arm- -chair chair � � Other tubes are called Other tubes are called chiral chiral and have angles and have angles � between 0 and 30 between 0 and 30 � Figure (fig 4 ,5) shows these three structures Figure (fig 4 ,5) shows these three structures � and STM image of a SWNT and STM image of a SWNT
Electronic Properties Electronic Properties
Electronic properties Electronic properties � Electronic structure of Electronic structure of Graphene Graphene � � In In graphene graphene, a bonding , a bonding π π - -band and an anti band and an anti- -binding binding π π * *- - � band is formed band is formed � Wallace derived an expression for the 2 Wallace derived an expression for the 2- -D energy sates, D energy sates, � W2D, of the π π electrons as a function of wave vectors electrons as a function of wave vectors W2D, of the k x ,k y : k x ,k y : k a k a 3 k a = ± γ + + y y 2 1 / 2 x W ( k , k ) [ 1 4 cos( ) cos( ) 4 cos ( )] 2 D x y 0 2 2 2 � γ γ 0 denotes nearest neighbor overlap integral and 0 denotes nearest neighbor overlap integral and � a=0.246 nm is the in plane lattice constant a=0.246 nm is the in plane lattice constant � The two signs in the relation represent The two signs in the relation represent π π and and π π * *- -band band �
Electronic Structure of Graphene Graphene Electronic Structure of � Figure (fig 6) shows that Figure (fig 6) shows that π π and and π π * *- -band just band just � touch each other at the corners of the 2- -D D touch each other at the corners of the 2 Brillouin zone zone Brillouin
Electronic Structure of Graphene Graphene Electronic Structure of � In the vicinity of In the vicinity of Γ Γ point, the dispersion relation point, the dispersion relation � is parabolically parabolically shaped, while towards the shaped, while towards the is corners (K points) it shows a linear corners (K points) it shows a linear dependence on W(k W(k) ) dependence on � No energy gap exist in the No energy gap exist in the graphene graphene � dispersion relation, we are dealing with a dispersion relation, we are dealing with a gapless semiconductor gapless semiconductor � Real graphite is a metal and the bands Real graphite is a metal and the bands � overlap by 40 meV meV due to interaction of due to interaction of overlap by 40 graphene planes planes graphene
Electronic Structure of Carbon Electronic Structure of Carbon Nanotubes Nanotubes � For For CNTs CNTs, the structure is macroscopic along the tube , the structure is macroscopic along the tube � axis, but the circumference is in atomic scale axis, but the circumference is in atomic scale � Density of allowed quantum mechanical states in axial Density of allowed quantum mechanical states in axial � direction will be high, but the number of states in direction will be high, but the number of states in circumferential direction will be limited circumferential direction will be limited � Periodic boundary conditions will define allowed modes Periodic boundary conditions will define allowed modes � (1- -D states) along the tube axis according to: D states) along the tube axis according to: (1 = π = C . k 2 j with j 0,1,2, ... h For arm - chair tube s, allowed values for circumfere ntial direction are (based on periodic boundary conditions ) π j 2 = = = k , q m n y , j y q 3 a y
Electronic Structure of Carbon Nanotubes Nanotubes Electronic Structure of Carbon � Figure (fig 7) shows dispersion relation, the Figure (fig 7) shows dispersion relation, the � projection of allowed 1- -D states onto the first D states onto the first projection of allowed 1 Brillouin zone of zone of graphene graphene and and W(kx W(kx) relation ) relation Brillouin for a (3,3) tube for a (3,3) tube
Electronic Structure of Carbon Electronic Structure of Carbon Nanotubes Nanotubes � Allowed states condense into lines (there are Allowed states condense into lines (there are � qy=3 lines on either side of the center of the =3 lines on either side of the center of the qy Brillouin zone) zone) Brillouin � In case of (3,3) tube (and all other arm In case of (3,3) tube (and all other arm- -chair chair � tubes), the allowed states (lines) include the K tubes), the allowed states (lines) include the K points of Brillouin Brillouin zone of zone of graphene graphene, hence all , hence all points of arm- -chair tubes show a metallic behavior chair tubes show a metallic behavior arm
Electronic Structure of Carbon Nanotubes Nanotubes Electronic Structure of Carbon � Figure (fig 8) shows the dispersion relation, Figure (fig 8) shows the dispersion relation, � the projection of allowed 1- -D states onto D states onto the projection of allowed 1 Brillouin zone of zone of graphene graphene and the and the W(kx W(kx) ) Brillouin relation for a chiral chiral (4,2) tube (4,2) tube relation for a
Electronic Structure of Carbon Nanotubes Nanotubes Electronic Structure of Carbon � Ch vector is not parallel to y direction and Ch vector is not parallel to y direction and � there is a mixed quantization of kx kx and and ky ky there is a mixed quantization of � There are no modes which include the K There are no modes which include the K � points of the Brillouin Brillouin zone of zone of graphene graphene, WF is , WF is points of the now in a bandgap bandgap, therefore, this type of tube , therefore, this type of tube now in a is semiconductor with bandgap bandgap of few of few eV eV is semiconductor with � In general, In general, bandgap bandgap decreases with decreases with � increasing diameter of the tube increasing diameter of the tube
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