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- 1. Electronic Band Structure of Solids Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html 1
- 2. What are quantum numbers? Quantum numbers label eigenenergies and eigenfunctions of a Hamiltonian Sommerfeld: k -vector ( hk is momentum) Bloch: k -vector ( hk is the crystal momentum) and n (the band index).•The Crystal Momentum is not the Momentum of a Bloch electron: the rate of change of anelectron momentum is given by the total forces on the electron, but the rate of change ofelectronic crystal momentum is: dr 1 ∂ε n (k ) d hk = v n (k ) = Ψ nk v Ψ nk = ˆ ; = − e [ E(r, t ) + v n (k ) × B(r, t ) ] dt h ∂k dtwhere forces are exerted only by the external fields, and not by the periodic field of the lattice. PHYS 624: Electronic Band Structure of Solids 2
- 3. Semiclassical dynamics of Bloch electrons •Bloch states have the property that their expectation values of r and k , follow classical dynamics. The only change is that now εn (k ) (band structure) must be used: e H classical = εn [ hk + eA (r ) ] − eϕ(r ) + B× k L 2m dp d ( hk ) ∂H dr ∂H ∂ε = =− , =v = = dt dt ∂r dt ∂p ∂p •A perfectly periodic ionic arrangement has zero resistance. Resistivity comes from imperfections (example: a barrier induces a reflected and transmitted Bloch wave), which control the mean-free path. This can be much larger than the lattice spacing. •A fully occupied band does not contribute to the current since the electrons cannot be promoted to other empty states with higher k . The current is induced by rearrangement of states near the Fermi energy in a partially occupied band. n = const (no interband transitions) •Limits of validity: ε gap (k ) 2 eB ε gap (k ) 2 eEa = , hωc = = , ε gap (k ) = ε n (k ) − ε n′ (k ) εF m εF hω =?ε gap , λ a PHYS 624: Electronic Band Structure of Solids 3
- 4. What is the range of quantum numbers? Sommerfeld: k runs through all of k-space consistent with the Born-von Karman periodic boundary conditions: Ψ ( x + L, y , z ) = Ψ ( x, y , z ) 2π Ψ ( x + L, y , z ) = Ψ ( x, y + L , z ) ⇒ k = L ( nx , ny , nz ) ↔ nx , ny , nz = 0, ±1, ±2,K Ψ ( x, y , z ) = Ψ ( x , y , z + L ) Ψ ( x + L, y , z ) = Ψ ( x , y , z ) = 0 π Ψ ( x + L, y, z ) = Ψ ( x, y + L, z ) = 0 ⇒ k = ( nx , n y , nz ) ↔ nx , n y , nz = 1, 2,K L Ψ ( x, y , z ) = Ψ ( x , y , z + L ) = 0 Bloch: For each n, k runs through all wave vectors in a single primitive cell of the reciprocal lattice consistent with the Born- von Karman periodic boundary conditions; n runs through an infinite set of discrete values. PHYS 624: Electronic Band Structure of Solids 4
- 5. What are the energy levels? Sommerfeld: 2 2 hk ε (k ) = 2m Bloch: For a given band index n, ε n (k ) has no simple explicit form. The only general property is periodicity in the reciprocal space: ε n (k + G ) = ε n (k ) PHYS 624: Electronic Band Structure of Solids 5
- 6. What is the velocity of electron? Sommerfeld: The mean velocity of an electron in a level with wave vector k is: hk 1 ∂ε v= = m h ∂k Bloch: The mean velocity of an electron in a level with band index n and wave vector k is: 1 ∂ε n (k ) Conductivity of a perfect crystal: v n (k ) = σ →∞ h ∂k NOTE: Quantum mechanical definition of a mean velocity h∇ v ≡ Ψ v Ψ = ∫dr Ψ (r ) ˆ * Ψ r) ( mi PHYS 624: Electronic Band Structure of Solids 6
- 7. What is the Wave function Sommerfeld: The wave function of an electron with wave vector kis: 1 kΨ (r ) = e ikr V Bloch: The wave function of an electron with band index n and wave vector k is: Ψk (r ) = e ϑnk (r ) ikr where the function ϑnk ( r ) has no simple explicit form. The only general property is its periodicity in the direct lattice (i.e., real space): ϑnk (r + R ) = ϑnk (r ) PHYS 624: Electronic Band Structure of Solids 7
- 8. Sommerfeld vs. Bloch: Density of States Sommerfeld → Bloch 2 2 D(ε ) = ( 2π ) d∫ dk δ ( ε − ε (k ) ) → D(ε ) = ( 2π ) ∑ ∫ dk δ ( ε − ε (k ) ) d n B.Z . n PHYS 624: Electronic Band Structure of Solids 8
- 9. Bloch: van Hove singularities in theDOS 2 2 dS D(ε ) = ∑∫ dk δ ( ε − ε n (k ) ) = ∑ ∫ dk ∇ ε (k ) ( 2π ) ( 2π ) d d n B.Z . n Sn ( E ) k n 1 D(ε )d ε = ( 2π ) ∫ dS ∆k d d ε = ( ∇ k ε n (kΔk )) = ∇ k εk( ) ∆ k n D(ε ) PHYS 624: Electronic Band Structure of Solids 9
- 10. Bloch: van Hove singularities in theDOS of Tight-Binding Hamiltonianε n (k ) = − 2t ( cos(k x a) + cos(k y a) + cos(k z a) ) ⇒ ∇ k ε n (k ) = 2ta ( sin(k x a) + sin( k y a) + sin( k z a) ) k = (0, 0, 0) max π π π ∇k ε n (k ) = 0 for k = ± , ± , ± ÷min 1D a π a a π π k = ± , 0, 0 ÷, 0, ± , 0 ÷, 0, 0, ± ÷saddle a a a 3D Local DOS: ρ (r, ε ) = ∑ Ψα ( r ) δ (ε − ε α ) 2 α DOS: D(ε ) = ∫ dr ρ (r, ε ) PHYS 624: Electronic Band Structure of Solids 10
- 11. Sommerfeld vs. Bloch: Fermi surface•Fermi energy ε F = µ (T = 0) represents the sharp occupancycut-off at T=0 for particles described by the Fermi-Dirac statitics.•Fermi surface is the locus of points in reciprocal space where ε (k ) = ε F εF No Fermi surface for insulators! − kF + kF Points of Fermi “Surface” in 1D PHYS 624: Electronic Band Structure of Solids 11
- 12. Sommerfeld vs. Bloch: Fermi surface in 3D Sommerfeld: Fermi Sphere Bloch: Sometimes sphere, but more likely anything else For each partially filled band there will be a surface reciprocal space separating occupied from the unoccupied levels → the set of all such surfaces is known as the Fermi surface and represents the generalization to Bloch electrons of the free electron Fermi sphere. The parts of the Fermi surface arising from individual partially filled bands are branches of the Fermi surface: for each n solve the equation ε n (k ) = ε F in kvariable. PHYS 624: Electronic Band Structure of Solids 12
- 13. Is there a Fermi energy of intrinsic Semiconductors? •If ε F is defined as the energy separating the highest occupied from the lowest unoccupied level, then it is not uniquely specified in a solid with an energy gap, since any energy in the gap meets this test. •People nevertheless speak of “the Fermi energy” on an intrinsic semiconductor. What they mean is the chemical potential, which is well defined at any non-zero temperature. As T → 0 , the chemical potential of a solid with an energy gap approaches the energy of the middle of the gap and one sometimes finds it asserted that this is the “Fermi energy”. With either the correct of colloquial definition, ε n (k ) = ε F does not have a solution in a solid with a gap, which therefore has no Fermi surface! PHYS 624: Electronic Band Structure of Solids 13
- 14. DOS of real materials: Silicon, Aluminum, Silver PHYS 624: Electronic Band Structure of Solids 14
- 15. Colloquial Semiconductor “Terminology” in Pictures ←PURE DOPPED→ PHYS 624: Electronic Band Structure of Solids 15
- 16. Measuring DOS: Photoemissionspectroscopy Fermi Golden Rule: Probability per unit time of an electron being ejected is proportional to the DOS of occupied electronic states times the probability (Fermi function) that the state is occupied: 1 I (ε kin ) = µ D (−ε bin ) f (−ε bin ) µ D (ε kin + φ − hω ) f (ε kin + φ − hω ) τ (ε kin ) PHYS 624: Electronic Band Structure of Solids 16
- 17. Measuring DOS: Photoemissionspectroscopy Once the background is subtracted off, the subtracted data is proportional to electronic density of states convolved with a Fermi functions. We can also learn about DOS above the Fermi surface using Inverse Photoemission where electron beam is focused on the surface and the outgoing flux of photons is measured. PHYS 624: Electronic Band Structure of Solids 17
- 18. Fourier analysis of systems living onperiodic lattice f (r ) = f (r + R ) ⇒ f (r ) = ∑ f G e iGr 1 G ∫ ∫ −iGr fG = dr e f (r ); e iGr dr = 0 V pcell pcell pcell φ(k ) =φ(k +G ) ⇒φ(k ) = ∑ R eiRk φ R dk φR =V pcell BZ ∫ ( 2π ) 3 e −iRkφ(k ) Born-von Karman: f (r ) = f (r + N i ai ) ⇒ 3 mi 1 f (r ) = ∑ f k e ikr , k =∑ bi ; fk = ∫ dr e −ikr f (r ); ∫ eikr dr = 0 k i =1 Ni Vcrystal crystal crystal ∑e ikR = N δk ,0 3 R =∑ i ai , 0 ≤ni < N i , N = N1 N 2 N 3 ; k ∈BZ R n ∑e k ikR = N δR ,0 i=1 PHYS 624: Electronic Band Structure of Solids 18
- 19. Fouirer analysis of Schrödinger equation U (r ) = U (r + R ) ⇒U (r ) = ∑U G eiGr , G × = 2π m R G h2 2 ˆ Ψ(r ) ≡ − H 2m ∇ + U (r ) Ψ (r ) = εΨ (r ), Ψ (r ) = ∑ Ck eikr k h2 k 2 ∑ 2m k Ck e + ∑Ck ′U G e ikr k ′G i ( k ′+G)r = ε ∑Ck eikr k h2k 2 k′ → k − G : ∑ e ikr − ε ÷Ck + ∑ U G Ck-G = 0, for all r k 2m G h2k 2 − ε ÷Ck + ∑ U G Ck-G = 0, for all k ⇒ ε k = ε (k ) 2m G Potential acts to couple Ckwith its reciprocal space translation Ck +G and the problem decouples into N independent problems for each k within the first BZ. PHYS 624: Electronic Band Structure of Solids 19
- 20. Fourier analysis, Bloch theorem, andits corollaries − iGr ikr Ψ k (r ) = ∑ Ck −G e i ( k −G ) r = ∑ Ck −G e ÷e G G 744 644 ↓ 8 ϑ (r ) = ϑ (r + R ) ˆ H Ψk +G (r ) = ε (k + G )Ψ k +G (r ) 1. Ψ k +G (r ) = Ψk (r ) ⇒ ˆ H Ψk (r ) = ε (k + G )Ψ k (r ) → ε (k ) = ε (k + G ) h2 ( ∇ + ik ) + U (r) ϑnk (r) = ε n (k )ϑnk (r) 22. − •Each zone n is indexed by a k 2m vector and, therefore, has as many energy levels as there h2 are distinct k vector values ( ∇ − ik ) + U (r) ϑn,−k (r) = ε n (k )ϑn,−k (r) 23. − 2m within the Brillouin zone, i.e.: 4. ϑn ,k (r ) = ϑn ,− k (r ) , ε n (k ) = ε n ( −k ) (Kramers theorem) * N = N1 N 2 N3 PHYS 624: Electronic Band Structure of Solids 20
- 21. “Free” Bloch electrons? h2k 2 •Really free electrons → Sommerfeld ε n (k ) = continuous spectrum with infinitely degenerate eigenvalues. 2m • ε n (k )=ε n (k + G ) does not mean that two electrons with wave vectors k and k + G ε ( have the same energy, but that any reciprocal lattice point can serve as the originnofk ) . •In the case of an infinitesimally small periodic potential there is periodicity, but not a real potential. The ε n (k ) function than is practically the same as in the case of free electrons, but starting at every point in reciprocal space. Bloch electrons in the limit U →0 : electron moving through an empty lattice! PHYS 624: Electronic Band Structure of Solids 21
- 22. Schrödinger equation for “free” Blochelectronsh 2 0 h2 2m ( ∇ + ik ) + ε (k ) θ k (r ) = 0 ε (k ) = 2m ( k + G ) 2 0 0 2 1 iGr Counting of Quantum States: ϑk (r ) = 0 e V Extended Zone Scheme: FixG (i.e., the BZ) and then count k vectors within the region corresponding to that zone. Reduced Zone Scheme: Fix k in any zone and then, by changing G , count all equivalent states in all BZ. PHYS 624: Electronic Band Structure of Solids 22
- 23. “Free” Bloch electrons at BZ boundary πx πx Ψ + : ( eiGx / 2 + e −iGx / 2 ) : cos Ψ− : ( eiGx / 2 − e −iGx / 2 ) : sin a a PHYS 624: Electronic Band Structure of Solids 23
- 24. “Free” Bloch electrons at BZ boundary •Second order perturbation theory, in crystalline potential, for the reduced zone scheme: 2 H+k U G+kε k (G ) = ε (G ) + G + k U G + k + ∑ 0 k + ... + O(U 3 ) H ε k0 (G ) − ε k0 (H) 1 i (G +k)r r G + k = ϑ (r )e 0 nk ikr = e V 1 G + k U H + k = ∑ U G′ ∫ e i ( − H+G+G ′ ) dr = ∑ U G′δ G ′,G-H = U G-H V G′ G′ 1 G + k U H + k = ∫ U (r )dr V U G′ 2 second order correction = ∑ 0 G′ ε k (G ) − ε k (G ′ + G ) 0 PHYS 624: Electronic Band Structure of Solids 24
- 25. “Free” Bloch electrons at BZ boundary •For perturbation theory to work, matrix elements of crystal potential have to be smaller than the level spacing of unperturbed electron → Does not hold at the BZ boundary! 2 2 ε (G = 0) − ε (G ′) = 0 h 2 0 k − ( G′ + k ) 2 = h ( − G ′ 2 − 2G ′k ) 2m 2m k k 0 at BZ boundary ↔ Laue diffraction! Ψ = α eikr + β ei (G+k)r 1 0 (ε (k ) − ε (k ) ) + 4 U G′ 2 2 ε ± G=0 (k ) = ε G = 0 (k ) − ε G′ (k ) ± 0 0 G=0 0 G′ 2 G G G k = ⇒ ε G = 0 (k ) = ε G = 0 (k = ) + U G′ ; ε G = 0 (k ) = ε G = 0 (k = ) − U G′ + 0 − 0 2 2 2 PHYS 624: Electronic Band Structure of Solids 25
- 26. Extended vs. Reduced vs. RepeatedZone Scheme •In 1D model, there is always a gap at the Brillouin zone boundaries, even for an arbitrarily weak potential. •In higher dimension, where the Brillouin zone boundary is a line (in 2D) or a surface (in 3D), rather than just two points as here, appearance of an energy gap depends on the strength of the periodic potential compared with the width of the unperturbed band. PHYS 624: Electronic Band Structure of Solids 26
- 27. Fermi surface in 2D for free Sommerfeld electrons 2 2π 4π 2 S BZ =b = 2 ÷ = , Scell = a 2 a Scell S BZ 4π 2 2π 2 S state = = = , N = N1 N 2 2 N 2 NScell Scrystall 2π 2 S BZ Sb π S Fermi = N e S state = 2 = ⇒ kF = = 0.798 a 2 2π a π S Fermi = 2 N e S state = S BZ ⇒ k F = 1.128 a S BZ S Fermi = zN e S state =z , for crystal with z-valence atoms 2 PHYS 624: Electronic Band Structure of Solids 27
- 28. Fermi surface in 2D for “free” Bloch electrons •There are empty states in the first BZ and occupied states in the second BZ. •This is a general feature in 2D and 3D: Because of the band overlap, solid can be metallic even when if it has two electrons per unit cell. PHYS 624: Electronic Band Structure of Solids 28
- 29. Fermi surface is orthogonal to the BZ boundary ( ε 0 (k ) − ε 0 (k + G ) ) 2 ε (k ) + ε (k + G ) 0 0 2 ε (k ) = ± + UG 2 4 1 (ε (k ) − ε 0 (k + G ) ) ( ∇ k ε 0 (k ) − ∇ k ε 0 (k + G ) ) 0 1 ∇ k ε (k ) = ( ∇ k ε 0 (k ) + ∇ k ε 0 ( k + G ) ) ± 2 2 2 ( ε 0 (k ) − ε 0 (k + G) ) 2 2 + UG 4 at the BZ boundary : ε 0 (k ) = ε 0 (k + G ) h2 h2 k ∇ k ε (k ) = ( k + k + G ) ¬ + G ÷×G = 0 2m m2 PHYS 624: Electronic Band Structure of Solids 29
- 30. Tight-binding approximation →Tight Binding approach is completely opposite to “free” Bloch electron: Ignore core electron dynamics and treat only valence orbitals localized in ionic core potential. There is another way to generate band gaps in the electronic DOS → they naturally emerge when perturbing around the atomic limit. As we bring more atoms together or bring the atoms in the lattice closer together, bands form from mixing of the orbital states. If the band broadening is small enough, gaps remain between the bands. PHYS 624: Electronic Band Structure of Solids 30
- 31. Constructing Bloch functions fromatomic orbitals PHYS 624: Electronic Band Structure of Solids 31
- 32. From localized orbitals to wave functions overlap PHYS 624: Electronic Band Structure of Solids 32
- 33. Tight-binding method for single s-band →Tight Binding approach is completely opposite to “free” Bloch electron: Ignore core electron dynamics and treat only valence orbitals localized in ionic core potential. Notation: x i = ϕ( x − R i ) PHYS 624: Electronic Band Structure of Solids 33
- 34. One-dimensional case →Assuming that only nearest neighbor orbitals overlap: PHYS 624: Electronic Band Structure of Solids 34
- 35. One-dimensional examples:s-orbital band vs. p-orbital band tp >0 ts < 0 ts < t p +∞ Ψ kl ( x) = ∑ n = −∞ eikxφ l ( x − na ) ¬ Re [ Ψ kl ( x)] → bandwidth: W = 4dtl = 2 ztl PHYS 624: Electronic Band Structure of Solids 35
- 36. Wannier Functions →It would be advantageous to have at our disposal localized wave functions with vanishing overlap i j = δij : Construct Wannier functions as a Fourier transform of Bloch wave functions! PHYS 624: Electronic Band Structure of Solids 36
- 37. Wannier functions as orthormal basis set Ri − a Ri + a 1D example: decay as power law, so it is not completely localized! PHYS 624: Electronic Band Structure of Solids 37
- 38. Band theory of Graphite and Carbon Nanotubes(works also for MgB2 ): Application of TBH method •Graphite is a 2D network made of 3D carbon atoms. It is very stable material (highest melting temperature known, more stable than diamond). It peels easily in layers (remember pencils?). •A single free standing layer would be hard to peel off, but if it could be done, no doubt it would be quite stable except at the edges – carbon nanotubes are just this, layers of graphite which solve the edge problem by curling into closed cylinders. •CNT come in ‘’single-walled” and “multi-walled” forms, with quantized circumference of many sizes, and with quantized helical pitch of many types. Lattice structure of graphite layer: There are two carbon atoms per cell, r designated as the A and B sublattices. The vector τ connects the two r r sublattices and is not a translation vector. Primitive translation vectors are a, b . PHYS 624: Electronic Band Structure of Solids 38
- 39. Chemistry of Graphite: sp 2 hybridization, covalentbonds, and all of that 1 A, B 2 A, B φ1A, B = s ± px 3 3 1 A , B 1 A , B 1 A, B φ2A, B = s m px ± py 3 6 2 1 A, B 1 A, B 1 A, B φ3A, B = s ± px ± py 3 6 2 φ4A, B = pzA, B 1 A 1 A Ψ b ( onding ) i = ( φi +φi ) , Ψi B a ( ntibonding ) = ( φi −φiB ) 2 2 PHYS 624: Electronic Band Structure of Solids 39
- 40. Truncating the basis to a single π orbital per atom •The atomic s orbitals as well as the Eigenstates of translation operator: p ,p atomic carbon x y functions form strong bonding orbitals which are doubly occupied 1 ∑ and lie below the Fermi energy. They also form strongly antibonding orbitals which are kA = eikmφA (r − m) high up and empty. N m •This leaves space on energy axis near the 1 Fermi level for π orbitals (they point perpendicular to the direction of the bond kB = N m ∑ eikmφB (r − m ) between them) φB (r ) = φA (rτ ) − •The π orbitals form two bands, one bonding band lower in energy which is doubly occupied, and one antibonding band Bloch eigenstates: higher in energy which is unoccupied. •These two bands are not separated by a kn = α kA + β kB gap, but have tendency to overlap by a 2 2 small amount leading to a “semimetal”. α + β =1 PHYS 624: Electronic Band Structure of Solids 40
- 41. Diagonalize 2 x 2 Hamiltonian ˆ kA H kA ˆ kA H kB H (k ) = ÷ kB H kA ˆ ˆ kB ÷ kB H 1 kA H kA = ∑ eik ( m-m′) ∫ drφA (r − m′) HφA (r − m ) ˆ ˆ N mm′ kB H kB = ∫ drφB (r ) HφB (r ) = kA H kA = ∫ drφA (r ) HφA (r ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ kA H kB = ∫ drφA (r ) HφB (r ) + e −ika ∫ drφA (r ) HφB (r + a) + e −ikb ∫ drφA (r ) HφB (r + b) −t ε ± (k ) = ±t 1 + e −ika + e−ikb ε ± (k ) = ±t [ 3 + 2 cos(ka) + 2 cos(kb) + 2 cos(k (a − b)) ] 1/ 2 PHYS 624: Electronic Band Structure of Solids 41
- 42. Band structure plotting: Irreducible BZ Ψ (r ) = { R | Ta } Ψ (r ) = Ψ ({ R | Ta } −1 n Rk n k n k r ) = Ψk ( R −1r − R −1a) n = { R | −R a} −1 { R | Ta } r = Rr + a, { R | Ta } −1 −1 −1 PHYS 624: Electronic Band Structure of Solids 42
- 43. Graphite band structure in pictures •Plot ε (k ) for some special directions in reciprocal space: there are three directions of special symmetry which outline the “irreducible wedge” of the Brillouin zone. Any other point k of the zone which is not in this wedge can be rotated into a k-vector inside the wedge by a symmetry operation that leaves the crystal invariant. along Σ : ε ± (k ) = ±t 5 + 4 cos ( 2πς ) ( 0 < ς < 1) 1/ 2 middle section : ε ± (k ) = ±t 3 − 4 cos ( 2πς / 3 ) + 2 cos ( 4πς / 3 ) 1/ 2 along Λ : ε ± (k ) = ±t 3 + 2 cos ( 4πς / 3 ) + 4 cos ( 2πς / 3 ) 1/ 2 PHYS 624: Electronic Band Structure of Solids 43
- 44. Graphite band structure in pictures:Pseudo-Potential Plane Wave Method Electronic Charge Density: In the plane of atoms In the plane perpendicular to atoms PHYS 624: Electronic Band Structure of Solids 44
- 45. Diamond vs. Graphite: Insulator vs. Semimetal PHYS 624: Electronic Band Structure of Solids 45
- 46. Carbon Nanotubes •Mechanics: Tubes as ultimate fibers. •Electronics: Tubes as quantum wires. •Capillary: Tubes as nanocontainers. PHYS 624: Electronic Band Structure of Solids 46
- 47. From graphite sheets to CNT C-C distance: a = 1.421 A 1 3 primitive vectors: a1 = (1,0) a 2 = , ÷ 2 2 ÷ chiral vector: c h = n1a1 + n2a 2 circumference: L = a 3(n12 + n1n2 + n2 ) 2 d is highest common divisor of (2n1 + n2 , 2 n2 + n1 ) translation vector of 1D unit cell along the axis: R = t1a1 + t2a 2 3L modulues of translation vector: R = d 4(n12 + n1n2 + n22 ) •Single-wall CNT consists of rolling the number of atoms per unit cell: N = honeycomb sheet of carbon atoms into d a cylinder whose chirality and the fiber 3n2 diameter are uniquely specified by the chiral angle: θ = arctan vector: c = n a +n a 2n1 + n2 h 1 1 2 2 PHYS 624: Electronic Band Structure of Solids 47
- 48. Metallic vs. Semiconductor CNT The 1D band on CNT is obtained by slicing the 2D energy dispersion relation of the graphite sheet with the periodic boundary conditions: chk = 2π m ⇒ c h K ± = 2π m ⇔ 2n1 + n2 = 3m Conclusion: •The armchair CNT n1 = n2 are metallic •The chiral CNT with 2n1 + n2 ≠ 3m are moderate band-gap semiconductors. Metallic 1D energy bands are generally unstable under a Peierls distortion → CNT are exception since their tubular structure impedes this effects making their metallic properties at the level of a single molecule rather unique! PHYS 624: Electronic Band Structure of Solids 48
- 49. CNT Band structure PHYS 624: Electronic Band Structure of Solids 49

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