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", "Subsubsection"], Cell[CellGroupData[{ Cell["Turn off unneeded error messages first", "SmallText"], Cell[BoxData[ \(Off[General::spell1, Graphics::realu]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Lines connecting the nearest neighbour atoms:", "Text"], Cell[BoxData[ \(Clear[makelines]; \ \n makelines[l_, range_] := Module[{lines, n, i, j, dist}, \n\t\tlines = {}; \n\t\tn = Length[l]; \n \t\tFor[i = 1, i <= n, \(i++\), \n\t\t\t \(For[j = 1, j < i, \(j++\), \n\t\t\t\t dist = Sqrt[ \((l[\([i]\)] - l[\([j]\)])\) . \((l[\([i]\)] - l[\([j]\)])\)]; \n\t\t If[dist\ < \ range, \ \ \t \(lines = Append[lines, \ \ \ Line[{\ \ l[\([i]\)], l[\([j]\)]\ }]\ \ \ ]; \)\n \t\t\t\t\t]; \t\t\t\t\t\n\t\t\t]; \)\n\t\t]; \n\t\t Return[lines]]\)], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "Section", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell["\<\ Carbon nanotubes are huge tube-like molecules consisting only of \ Carbon atoms with a diameter in the nanometer range and lengths up to \ microns. The atoms are sitting in a hexagonal arrangement with 3-fold coordination (3 \ nearest neighbours). The nanotube can be viewed as a rolled up sheet of Graphene i.e. a 2 dimensional hexagonal lattice of Carbon atoms.\ \>", "Text"], Cell["The interatomic distance in \[CapitalARing]:", "Text"], Cell[BoxData[ \(\(d = 1.42; \)\)], "Input"], Cell["\<\ Unit vectors of the hexagonal lattice (like in a sheet of graphite \ = graphene) lying in the xy-plane:\ \>", "Text"], Cell["\<\ e1=(2*d*Cos[Pi/6])*{Cos[-Pi/6],Sin[-Pi/6],0}; e2=(2*d*Cos[Pi/6])*{Cos[Pi/6],Sin[Pi/6],0}; \ \>", "Input", PageWidth->Infinity], Cell["\<\ The atoms in the basis of the hexagonal lattice are often called \ A-atoms and B-atoms. 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3dX601=81@0@BPD04dP5015::@440981000d0@03004101814dP00`010@090@03BP4100/14dP80@U: 2@4CB0L134X70@0304Q801581P4D1001@0@03 004102@100<00@40@`4306L1103X0@00D@400`010@0T0@03004103l10`1W0@<0k040058100<00@40 90400`010@0j0@@0IP440>l1001C0@03004102@100<00@40=P4306L10`3c0@00E04202D100<00@40 04006`100<00@406@400`010@1:0@<0 o`4l0@00K@400`010@0H0@03004104H1103o0Cl1001^0@03004101L100<00@40@`430?l1@`4006l1 0P0G0@03004103l1103o0DH1001a0@03004101@100<00@40?0430?l1BP40078100<00@404`400`01 0@0i0@<0o`5=0@00L`400`010@0B0@03004103D1103o0E01001d0@030041014100<00@40"], ImageRangeCache->{{{0, 530.938}, {296.312, 0}} -> {-0.00111066, 0.19073, 0.00188773, 0.00188773}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ The wrap up of the nanotube is done by identifying the A-atom at \ {0,0,0} with the A-atom at nn1*e1+nn2*e2, where nn1 and nn2 are integer numbers: These are indices which describe the \ nanotube geometry completely. The vector nn1*e1+nn2*e2 is the circumferential vector and the tube will be \ directed perpendicular to this. The structure formed will in general have chiral (translation along the tube \ axis + rotation around the tube axis) and rotational symmetry. These symmetries are used to classify the quantum \ energy eigenstates of the tubes below.\ \>", "Text"], Cell["\<\ The Carbon nanotubes are e.g. described in the paper by White & \ Mintmire, Carbon,Vol. 33, No. 7, pp. 893-902, 1995. In the notebook the \ notation will primarily be adapted from this paper. A nice introduction by C. \ Dekker can be found in Physics Today, May 1999 p. 22. The homepage of C. Dekker (http://vortex.tn.tudelft.nl/~dekker/) contains \ nice pictures from this latter reference. \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Atomic Structure", "Section", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell["\<\ nn1 and nn2 is the index defining the carbon nanotube (defining its \ chiral vector), u is a measure of the length of the piece we want to plot.\ \>", "Text"], Cell[BoxData[ \(Clear[makenanotube]; \n makenanotube[nn1_, nn2_, u_] := Module[\n\t\t{coors, i, j, theta, rotxy, E1, E2, NC, H, B, BE, h, phi, Radius, \n\t\t\tTA, Btmp, BEtmp, S, dbasis, hbasis, phibasis}, \n \t\t\n\t\tn1 = Min[nn1, nn2]; \n\t\tn2 = Max[nn1, nn2]; \n\n\t Btmp = \((n1*e1 + n2*e2)\); \n\t\tBEtmp = Btmp/Sqrt[Btmp . Btmp]; \n \t\t (*\ redefine\ e1, \ e2\ so\ B, BE\ is\ along\ the\ x - axis, \ \(--\ the\)\ Tube\ is\ along\ y\ *) \n\t\t theta = ArcCos[BEtmp . {1, 0, 0}]; \n\t\t rotxy = {{Cos[theta], Sin[theta], 0}, {\(-Sin[theta]\), Cos[theta], 0}, {0, 0, 1}}; \n\t\tE1 = rotxy . e1; \n\t\tE2 = rotxy . e2; \n \t\tB = {Sqrt[Btmp . Btmp], 0, 0}; \n\t\tBE = {1, 0, 0}; \n\t\t\n \t\t\tRadius = Sqrt[B . B]/\((2 Pi)\); \n\t\ \ Print["\", Radius]; \n\t\t\n \t\t (*\ Determine\ m1, m2\ *) \n\t\t\t\tNC = GCD[n1, n2]; \n \ \ \ \ \ \ \ \ \ Print["\", NC, "\<-order axis\>"]; \n \t\t\t\tm1 = 0.5; \n\t\tm2 = \(-1\); \ \n\t While[IntegerPart[m1] < m1, \n\t\t\tm2 = m2 + 1; \n\t\t\t m1 = \((1 + m2*\((n1/NC)\))\)/\((n2/NC)\); \n\t\t]; \n\t\t Print["\", m1, "\<,\>", m2]; \n\t\tH = E1*m1 + E2*m2; \n \t\th = Sqrt[\((Cross[H, B] . Cross[H, B])\)/\((B . B)\)]; \n\t\t phi = 2 Pi \((H . B)\)/\((B . B)\); \n\n\t\t\t\t\t Print["\<(h,phi) = (\>", h, "\<,\>", phi, "\<)\>"]; \n\n \t\t (*\ Chiral\ \(operator : \ translate\ h\ in\ y\ dir . \ and\ rotate\ phi\ around\ y\ axis \)\ *) \n\t S[r_] := \ \ {0, h, 0}\ + \((\ {{Cos[phi], 0, Sin[phi]}, {0, 1, 0}, {\(-Sin[phi]\), 0, Cos[phi]}} . r)\); \ \n\t\t\n \t\t (*\ Rotation\ operator\ Cn\ *) \n\t\t Cn[r_] = \ {{Cos[2 Pi/NC], 0, Sin[2 Pi/NC]}, {0, 1, 0}, { \(-Sin[2 Pi/NC]\), 0, Cos[2 Pi/NC]}} . r; \n\t\t\n \t\t (*\ unit - cell\ coordinates\ uc\ *) \n\t\n\t\t aatom = {Radius, 0, 0}; \ \n\t\tdbasis = \((E1 + E2)\)/3\ ; \n\t\t hbasis = Sqrt[\((Cross[dbasis, {1, 0, 0}] . Cross[dbasis, {1, 0, 0}])\)]; \n \t\tphibasis = 2 Pi*\((dbasis . B)\)/\((B . B)\); \n\t\t batom = \t\ {0, hbasis, 0}\ + \n\t\t\t\t\t \((\ {{Cos[phibasis], 0, Sin[phibasis]}, {0, 1, 0}, { \(-Sin[phibasis]\), 0, Cos[phibasis]}} . {Radius, 0, 0}) \); \ \n\t\t\n \t (*\ Generate\ coordinates\ using\ symmetry\ operations\ *) \n\t\t\n \t\tcoors = Flatten[\n\t\t\t\t Table[{Nest[Cn, Nest[S, aatom, i], j], Nest[Cn, Nest[S, batom, i], j]}, {i, 0, u}, {j, 1, NC}], 1]; \n\t\tReturn[coors]\n\n\t]\)], "Input"], Cell[CellGroupData[{ Cell["Plot it and make a small movie", "Subsection"], Cell["\<\ Note that the distances are not quite equal to the d value -- especially for small, highly curved tubes\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(ntcoor = makenanotube[5, 5, 10]; \)\)], "Input"], Cell[BoxData[ InterpretationBox[\("Radius = "\[InvisibleSpace]3.39000028785736917`\), SequenceForm[ "Radius = ", 3.3900002878573692], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ \("Rotational symmetry with "\[InvisibleSpace]5 \[InvisibleSpace]"-order axis"\), SequenceForm[ "Rotational symmetry with ", 5, "-order axis"], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ \("m1,m2 = "\[InvisibleSpace]1\[InvisibleSpace]","\[InvisibleSpace]0\), SequenceForm[ "m1,m2 = ", 1, ",", 0], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ \("(h,phi) = ("\[InvisibleSpace]1.22975607337390258` \[InvisibleSpace]","\[InvisibleSpace]0.628318530717958623` \[InvisibleSpace]")"\), SequenceForm[ "(h,phi) = (", 1.2297560733739026, ",", 0.62831853071795862, ")"], Editable->False]], "Print"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ nanotubeplot= Show[Graphics3D[{ {RGBColor[0,0,1],AbsolutePointSize[20], Point[#]}&/@Transpose[ntcoor][[1]], {RGBColor[1,0,0],AbsolutePointSize[20], Point[#]}&/@Transpose[ntcoor][[2]], {AbsoluteThickness[6],#}&/@(makelines[Flatten[ntcoor,1],1.1*d]) }], ViewPoint-> {-2.136, -0.058, 0.112} ] \ \>", "Input", PageWidth->Infinity, FontFamily->"Courier New", FontSize->12, FontWeight->"Bold", FontColor->GrayLevel[0], Background->GrayLevel[1]], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics3D \[SkeletonIndicator]\), False, Editable->False]], "Output"], Cell[BoxData[ \(\(cameraat = {ViewPoint -> {\(-4.999\), \ 0.048, \ 0.062}, ViewPoint -> {\(-4.786\), \ \(-0.146\), \ 1.441}, ViewPoint -> {\(-3.971\), \ \(-0.121\), \ 3.035}, ViewPoint -> {\(-2.733\), \ 0.026, \ 4.187}, ViewPoint -> {\(-0.986\), \ \(-0.010\), \ 4.902}, ViewPoint -> {\(-0.392\), \ \(-0.004\), \ 4.985}, ViewPoint -> {\(-0.353\), \ \(-0.004\), \ 4.988}, ViewPoint -> {\(-0.329\), \ 0.127, \ 4.988}, ViewPoint -> {\(-0.286\), \ 0.208, \ 4.988}, ViewPoint -> {\(-0.214\), \ 0.281, \ 4.988}, ViewPoint -> {\(-0.135\), \ 0.326, \ 4.988}, ViewPoint -> {\(-0.022\), \ 0.352, \ 4.988}, ViewPoint -> {0.093, \ 0.341, \ 4.988}, ViewPoint -> {0.203, \ 0.289, \ 4.988}, ViewPoint -> {0.292, \ 0.198, \ 4.988}, ViewPoint -> {0.346, \ 0.069, \ 4.988}, ViewPoint -> {0.353, \ 0.014, \ 4.988}, ViewPoint -> {1.529, \ \(-0.001\), \ 4.761}, ViewPoint -> {3.108, \ \(-0.002\), \ 3.917}, ViewPoint -> {4.597, \ \(-0.003\), \ 1.968}, ViewPoint -> {4.871, \ \(-0.004\), \ 1.129}, ViewPoint -> {4.998, \ \(-0.004\), \ \(-0.137\)}, ViewPoint -> {4.859, \ \(-0.692\), \ \(-0.135\)}, ViewPoint -> {4.338, \ \(-2.297\), \ \(-0.135\)}, ViewPoint -> {3.732, \ \(-3.188\), \ \(-0.135\)}, ViewPoint -> {3.039, \ \(-3.854\), \ \(-0.135\)}, ViewPoint -> {2.331, \ \(-4.319\), \ \(-0.135\)}, ViewPoint -> {1.295, \ \(-4.734\), \ \(-0.135\)}, ViewPoint -> {\(-0.116\), \ \(-4.907\), \ \(-0.135\)}, ViewPoint -> {\(-0.100\), \ \(-4.257\), \ \(-0.117\)}, ViewPoint -> {\(-0.093\), \ \(-3.957\), \ \(-0.109\)}, ViewPoint -> {\(-0.071\), \ \(-3.018\), \ \(-0.083\)}, ViewPoint -> {\(-0.059\), \ \(-2.518\), \ \(-0.069\)}, ViewPoint -> {\(-0.055\), \ \(-2.318\), \ \(-0.064\)}, ViewPoint -> {\(-0.046\), \ \(-1.949\), \ \(-0.054\)}, ViewPoint -> {\(-0.039\), \ \(-1.669\), \ \(-0.046\)}, ViewPoint -> {\(-0.036\), \ \(-1.519\), \ \(-0.042\)}, ViewPoint -> {\(-0.028\), \ \(-1.189\), \ \(-0.033\)}, ViewPoint -> {\(-0.024\), \ \(-1.009\), \ \(-0.028\)}, ViewPoint -> {\(-0.026\), \ \(-1.089\), \ \(-0.030\)}, ViewPoint -> {\(-0.027\), \ \(-1.149\), \ \(-0.032\)}, ViewPoint -> {\(-0.029\), \ \(-1.249\), \ \(-0.034\)}, ViewPoint -> {\(-0.033\), \ \(-1.389\), \ \(-0.038\)}, ViewPoint -> {\(-0.037\), \ \(-1.569\), \ \(-0.043\)}, ViewPoint -> {\(-0.041\), \ \(-1.759\), \ \(-0.048\)}, ViewPoint -> {\(-0.053\), \ \(-2.229\), \ \(-0.061\)}, ViewPoint -> {\(-0.068\), \ \(-2.878\), \ \(-0.079\)}, ViewPoint -> {\(-0.088\), \ \(-3.738\), \ \(-0.103\)}, ViewPoint -> {\(-0.122\), \ \(-5.187\), \ \(-0.143\)}, ViewPoint -> {\(-1.052\), \ \(-5.080\), \ \(-0.143\)}, ViewPoint -> {\(-2.848\), \ \(-4.336\), \ \(-0.143\)}, ViewPoint -> {\(-4.336\), \ \(-2.848\), \ \(-0.143\)}, ViewPoint -> {\(-4.971\), \ \(-1.487\), \ \(-0.143\)}, ViewPoint -> {\(-5.130\), \ \(-0.771\), \ \(-0.143\)}, ViewPoint -> {\(-5.188\), \ 0.000, \ \(-0.143\)}}; \)\)], "Input"], Cell[BoxData[""], "Input"], Cell[BoxData[ \(ntmovie = Table[Show[nanotubeplot, cameraat[\([i]\)]], {i, Length[cameraat]}]\)], "Input", AnimationDisplayTime->0.169847, AnimationCycleOffset->1, AnimationCycleRepetitions->Infinity] }, Open ]] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData["Electronic Structure (only \[Pi]-bands)"], "Section", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[TextData[ "The electronic quantum energy eigenstates of the nanotubes can be \ classified/indexed by a rotational quantum number \ncorresponding to the \ rotational symmetry of the nanotube and a chiral quantum number corresponding \ to the \nchiral symmetry. A nanotube will of cause also have translational \ symmetry but this symmetry is already contained in the (more complete) chiral \ symmetry. Below we will plot the eigenenergy versus chiral-symmetry index (\ \[Kappa]). We will also make the corresponding plot where we use the \ translational symmetry index - or k-point which is the 1-dimensional \ band-structure known from basic solid state physics textbooks."], "Text"], Cell[CellGroupData[{ Cell["Hamiltonian (in eV):", "Section"], Cell[BoxData[ \(\(Vpppi = \(-2.77\); \)\)], "Input"], Cell[BoxData[ \(\(ham[r1_, r2_] := Vpppi*If[\((r1 - r2)\) . \((r1 - r2)\)\ < \ \((1.1*d)\)^2, 1.0, 0.0]; \)\)], "Input"] }, Closed]], Cell[BoxData[ \(\(Clear[nanotubebands]; \n nanotubebands[nn1_, nn2_] := Module[\n\t\t{othercoors, i, j, theta, rotxy, E1, E2, NC, H, B, BE, h, phi, Radius, aatom, batom, m1, m2, n1, n2, \n\t\t\tBtmp, BEtmp, S, invS, Cn, dbasis, hbasis, phibasis, phaselist, \n\t\tbandplotkappa, bandplotky, allbandplots, HAB, M, kappa, NS, eig1, k1, kybands, T, ky}, \n\t\t\n\t\tn1 = Min[nn1, nn2]; \n\t\tn2 = Max[nn1, nn2]; \n\n \t\ \ Btmp = \((n1*e1 + n2*e2)\); \n\t\t BEtmp = Btmp/Sqrt[Btmp . Btmp]; \n\t\t\n \t\t (*\ redefine\ e1, \ e2\ so\ B, BE\ is\ along\ the\ x - axis, \ \(--\ the\)\ Tube\ is\ along\ y\ *) \n\t\t theta = ArcCos[BEtmp . {1, 0, 0}]; \n\t\t rotxy = {{Cos[theta], Sin[theta], 0}, {\(-Sin[theta]\), Cos[theta], 0}, {0, 0, 1}}; \n\t\tE1 = rotxy . e1; \n\t\tE2 = rotxy . e2; \n \t\tB = {Sqrt[Btmp . Btmp], 0, 0}; \n\t\tBE = {1, 0, 0}; \n\t\t\n \t\t\tRadius = Sqrt[B . B]/\((2 Pi)\); \n\t\ \ Print["\", Radius]; \n\t\t\n \t\t (*\ Determine\ m1, m2\ *) \n\t\t\t\tNC = GCD[n1, n2]; \n \ \ \ \ \ \ \ \ \ Print["\", NC, "\<-order axis\>"]; \n \t\t\t\tm1 = 0.5; \n\t\tm2 = \(-1\); \ \n\t While[IntegerPart[m1] < m1, \n\t\t\tm2 = m2 + 1; \n\t\t\t m1 = \((1 + m2*\((n1/NC)\))\)/\((n2/NC)\); \n\t\t]; \n\t\t Print["\", m1, "\<,\>", m2]; \n\t\tH = E1*m1 + E2*m2; \n \t\th = Sqrt[\((Cross[H, B] . Cross[H, B])\)/\((B . B)\)]; \n\t\t phi = 2 Pi \((H . B)\)/\((B . B)\); \n\t\t Print["\<(h,phi) = (\>", h, "\<,\>", phi, "\<)\>"]; \n\n \t\t (*\ Chiral\ \(operator : \ translate\ h\ in\ y\ dir . \ and\ rotate\ phi\ around\ y\ axis \)\ *) \n\t\t\t\tClear[S]; \n\t S[r_] := \ \ {0, h, 0}\ + \((\ {{Cos[phi], 0, Sin[phi]}, {0, 1, 0}, {\(-Sin[phi]\), 0, Cos[phi]}} . r)\); \ \n\t\t\t\n\t\t\t\tClear[invS]; \n\t\t invS[r_] := \ \ {0, \(-h\), 0}\ + \((\ {{Cos[\(-phi\)], 0, Sin[\(-phi\)]}, {0, 1, 0}, { \(-Sin[\(-phi\)]\), 0, Cos[\(-phi\)]}} . r)\); \n\t\t\n \t\t (*\ Rotation\ operator\ Cn\ *) \n\t\tClear[Cn]; \n\t\t Cn[r_] := \ {{Cos[2 Pi/NC], 0, Sin[2 Pi/NC]}, {0, 1, 0}, { \(-Sin[2 Pi/NC]\), 0, Cos[2 Pi/NC]}} . r; \n\t\t\n \t\t (*\ unit - cell\ coordinates\ uc\ *) \n\t\t aatom = {Radius, 0, 0}; \ \n\t\tdbasis = \((E1 + E2)\)/3\ ; \n\t\t hbasis = Sqrt[\((Cross[dbasis, {1, 0, 0}] . Cross[dbasis, {1, 0, 0}])\)]; \n \t\tphibasis = 2 Pi*\((dbasis . B)\)/\((B . B)\); \n\t\t batom = \t\ {0, hbasis, 0}\ + \n\t\t\t\t\t \((\ {{Cos[phibasis], 0, Sin[phibasis]}, {0, 1, 0}, { \(-Sin[phibasis]\), 0, Cos[phibasis]}} . {Radius, 0, 0}) \); \ \n\tPrint["\", phibasis]; \n\t\n \t\t (*\ nanotube\ y\ unit - cell\ *) \n\t\t dR = GCD[2*\((n1)\) + n2, 2*\((n2)\) + n1]; \n\t\t T = Sqrt[3]*2 Pi*Radius/dR; \n\t\t Print["\", T]; \t\n\t\t NT = Ceiling[T/h]; \n\t\t\t Print["\", NT, \ "\< chiral operations\>"]; \t\n \t (*\ Generate\ bands\ from\ coordinates\ using\ symmetry\ operations \ *) \n\t\t\n \t\t (*\ Note\ we\ use\ that\ we\ have\ chiral\ "\"\ symmetry\ in\ the\ sense\ *) \n \t\t (*\ that\ \((1/S)\)\ generates\ an\ equivalent\ atom \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \ \n\t\t \n\t\t (*\ We\ must\ find\ matrix\ elements\ which\ give\ non - zero\ value\ on\ the\ B\ basis\ \(state : \)\ *) \n \t\t (*\ We\ search\ among\ NS\ A - equivalent\ coordinates\ \((excl . \ A\ it\ self)\)\ *) \n\t\t NS = 2*\((n1 + n2)\); \n\t\t abhamlist = Flatten[\n \t\t\t\t{ Table[ham[aatom, Nest[Cn, Evaluate[Nest[invS, batom, i]], j]], { j, 0, NC - 1}, {i, 1, NS}], \n\ \ \ \ \ \ \ \ \ \ Table[ham[aatom, Nest[Cn, Evaluate[Nest[\ S, batom, i]], j]], { j, 0, NC - 1}, {i, 0, NS}]}\n\t\t\t\ \ \ \ \ \ \ \ \ ]; \n\t \n\t\t (*\ Table\ of\ bandstructure\ plots\ for\ different\ rotation\ \(symmetry : \ M\)\ *) \n\t\tClear[kappa]; \n\t\t allbandplots = Table[\t Print["\< ======= Bands for rotational quantum number M = \>", M, "\< ======= \>"]; \ \n\t\t\t\t phaselist[kappa_] = \n\t\t\t\t\t Flatten[{ Table[Exp[\(+I\)*kappa*i]*Exp[\(-2\) Pi*I*M*j/NC], {j, 0, NC - 1}, {i, 1, NS}], Table[Exp[\(-I\)*kappa*i]*Exp[\(-2\) Pi*I*M*j/NC], {j, 0, NC - 1}, {i, 0, NS}]}]; \n\t\t\t\t HAB[kappa_] = \((Flatten[phaselist[kappa]])\) . \((Flatten[abhamlist])\); \n \t\t\t\tbandplotkappa = Plot[{\(-Abs[HAB[kappa]]\), Abs[HAB[kappa]]}, {kappa, \(-Pi\), Pi}, PlotStyle -> {AbsoluteThickness[0.15]}, Frame -> True, FrameLabel -> {"\<\[Kappa]\>", "\<\[CurlyEpsilon]\>"}, \n \t\t\t\t\ \ \ FrameTicks -> {{\(-Pi\), \(-Pi\)/2, 0, Pi/2, Pi}, Automatic, {}, {}}]; \n\t\t\t\t\n ky[kappa_] = Mod[\((kappa)\)*T/h\ \ - \ \((2 Pi/NC)\)*M*\((\((phi*T/h)\)/\((2 Pi/NC)\))\), 2 Pi]; \t\n\t\t\t\t\n\t\t\t\t\t\t kybands = Table[\t\t\t\t\t\n\ \ \ \ \ \ \ \ \ \ eig1 = Abs[HAB[kappa]]; \ k1 = ky[kappa]; \n \t\ \ \ \ \ \ \ \t\t{{\(-k1\), \(-eig1\)}, {k1, \(-eig1\)}, { \(-k1\), eig1}, {k1, eig1}}, \n \t\ \ \ \ \ \ \ \t\t{kappa, \(-Pi\), Pi, NC*2 Pi/400}]; \ \n \t\t\t\tbandplotky = ListPlot[Flatten[kybands, 1], PlotStyle -> {AbsolutePointSize[3]}, \ \n\t\t\t\t\t\t PlotLabel -> M, Frame -> True, FrameLabel -> {\*"\"\<\!\(k\_y\)\>\"", "\<\[CurlyEpsilon]\>"}, PlotRange -> {{\(-Pi\), Pi}, Automatic}, \ \n\t\t\t\t\t FrameTicks -> {{\(-Pi\), \(-Pi\)/2, 0, Pi/2, Pi}, Automatic, {}, {}}]; \n \t\t\t\t\t{bandplotkappa, bandplotky}, {M, 0, NC - 1}]; \n\t\t Print["\"]; \n\t Show[\(Transpose[allbandplots]\)[\([1]\)]]; Show[\(Transpose[allbandplots]\)[\([2]\)]]; \n\t\t\t\t\t]\n\t\)\)], "Input"], Cell[CellGroupData[{ Cell["Example: (5,5) Tube", "Section"], Cell[BoxData[ \(nanotubebands[5, 5]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Example: (9,2) Tube", "Section"], Cell[BoxData[ \(nanotubebands[9, 2]\)], "Input"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ More detailed electronic structure calculation with sp-orbital basis using Slater-Koster matrix elements\ \>", "Section", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell["Hamiltonian", "Section"], Cell[CellGroupData[{ Cell["\<\ White & Mintmire's (see e.g. Carbon,Vol. 33, No. 7, pp. 893-902, \ 1995 for a review) parameters (in eV):\ \>", "Subsubsection"], Cell[BoxData[ \(es = \(-6. \); ep = 0. ; Vpppi = \(-2.77\); Vpps = 4.37; Vsss = \(-4.76\); \ Vsps = 4.33; \)], "Input"] }, Open ]], Cell["\<\ The order of orbitals is: s, px, py, pz\ \>", "Subsubtitle"], Cell[BoxData[{ FormBox[ RowBox[{ \(SKham\ calculates\ \(HAB\ : \ \ which\ is\ part\ of\ the\ atom\ A\), \ atom\ B\), " "}], TextForm], FormBox[ RowBox[{"\t", RowBox[{ RowBox[{ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {"HAA", "HAB"}, {"HBA", "HBB"} }], "\[NegativeThinSpace]", ")"}], " ", "hamiltonian"}], ",", "\n", " ", \(where\ A\ and\ B\ refers\ to\ the\ two\ basis\ atoms . \ Each\)}]}], TextForm], FormBox[ RowBox[{"\t\t ", RowBox[{ RowBox[{ \(block\ is\ \((number\ of\ orbitals)\)\ x\ \((number\ of\ orbitals)\)\), " ", "=", " ", RowBox[{ \(4\ x\ 4. \ If\ the\ distance\ does\ not\ fit\ a\ typical\ A - B\ distance\ it\ returns\ zero . \n\t\t\t\t\nHAB\), "=", RowBox[{ TagBox[ RowBox[{"(", GridBox[{ {"Vss\[Sigma]", \(l\ Vsp\[Sigma]\), \(m\ Vsp\[Sigma]\), \(n\ Vsp\[Sigma]\)}, {\(\(-l\)\ Vsp\[Sigma]\), \(\((1 - l\^2)\)\ Vpp\[Pi] + l\^2\ Vpp\[Sigma]\), \(l\ m\ \((\(-Vpp\[Pi]\) + Vpp\[Sigma])\)\), \(l\ n\ \((\(-Vpp\[Pi]\) + Vpp\[Sigma])\)\)}, {\(\(-m\)\ Vsp\[Sigma]\), \(l\ m\ \((\(-Vpp\[Pi]\) + Vpp\[Sigma])\)\), \(\((1 - m\^2)\)\ Vpp\[Pi] + m\^2\ Vpp\[Sigma]\), \(m\ n\ \((\(-Vpp\[Pi]\) + Vpp\[Sigma])\)\)}, {\(\(-n\)\ Vsp\[Sigma]\), \(l\ n\ \((\(-Vpp\[Pi]\) + Vpp\[Sigma])\)\), \(m\ n\ \((\(-Vpp\[Pi]\) + Vpp\[Sigma])\)\), \(\((1 - n\^2)\)\ Vpp\[Pi] + n\^2\ Vpp\[Sigma]\)} }], ")"}], (MatrixForm[ #]&)], " ", "using", " ", "px"}]}]}], ",", "py", ",", "pz"}], "\n"}], TextForm], FormBox[ \(However\ we\ intend\ to\ use\ the\ {s, pr, pl = py, pt}\ basis\ which\ is\ invariant\ with\ respect\ to\ the\ symmetry\ \(operations . \)\n\t | \(pr > \)\ = \ Cos \((\[CurlyPhi])\)\ | px > \ \(+\ \ Sin\) \((\[CurlyPhi])\)\ | \(pz > \), \ \ \t \(|\( pt > \)\)\ = \ \(-Sin\) \((\[CurlyPhi])\)\ | px > \ \(+\ \ Cos\) \((\[CurlyPhi])\)\ | \(px > \)\), TextForm]}], "Subsubtitle"], Cell[BoxData[ \(\(\nClear[SKHam, l, m, n]; \n SKHam[l_, m_, n_] = \t{\n\t\t\t\t{Vsss, l*Vsps, m*Vsps, n*Vsps}, \n \t\t\ {\(-l\)*Vsps, l*l*Vpps + \((1 - l*l)\)*Vpppi, l*m*\((Vpps - Vpppi)\), l*n*\((Vpps - Vpppi)\)}, \n \t\t\t{\(-m\)*Vsps, l*m*\((Vpps - Vpppi)\), m*m*Vpps + \((1 - m*m)\) Vpppi, m*n*\((Vpps - Vpppi)\)}, \n \t\t\t{\(-n\)*Vsps, l*n*\((Vpps - Vpppi)\), m*n*\((Vpps - Vpppi)\), n*n*Vpps + \((1 - n*n)\)*Vpppi}}; \)\)], "Input"], Cell[BoxData[ \(\(SKhamOnsite = DiagonalMatrix[{es, ep, ep, ep, es, ep, ep, ep}]; \)\)], "Input"], Cell[BoxData[ \(\(\nClear[SKhamHOP12]; \n SKhamHOP12[r1_, r2_] := Module[{R, l, m, n, p, r1xz, r2xz, mtrans1, mtrans2}, \n\t\t R = r1 - r2; \n\t\tLR = Sqrt[R . R]; \n\t\t If[LR\ > \ \((1.1*d)\), Return[DiagonalMatrix[{0, 0, 0, 0, 0, 0, 0, 0}]]]; \n\t\t\t If[LR\ < \ \((0.9*d)\), Return[DiagonalMatrix[{0, 0, 0, 0, 0, 0, 0, 0}]]]; \n\tUR = R/LR; \n \t\t{l, m, n} = UR; \t\n\t\tr1xz = {r1[\([1]\)], r1[\([3]\)]}; \n\t\t r1xz = r1xz/Sqrt[r1xz . r1xz]; \n\t\t r2xz = {r2[\([1]\)], r2[\([3]\)]}; \n\t\ \ r2xz = r2xz/Sqrt[r2xz . r2xz]; \n\t\t mtrans1 = {{1, 0, 0, 0}, {0, r1xz[\([1]\)], 0, \(-r1xz[\([2]\)]\)}, { 0, 0, 1, 0}, {0, r1xz[\([2]\)], 0, r1xz[\([1]\)]}}; \n\t\t mtrans2 = {{1, 0, 0, 0}, {0, r2xz[\([1]\)], 0, \(-r2xz[\([2]\)]\)}, { 0, 0, 1, 0}, {0, r2xz[\([2]\)], 0, r2xz[\([1]\)]}}; \n \t\t\t (*\ the\ order\ in\ the\ matrix\ is\ s, pr, pl = py, pt\ *) \n \t\tm12 = Transpose[mtrans2] . SKHam[l, m, n] . mtrans1; \n\t\t H12 = Transpose[{{1, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0}, { 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0}}] . m12 . {{0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0, 0, 1}}; \n\t\n Return[H12]\n\t\t]\n\n\)\)], "Input"] }, Open ]], Cell["\<\ nkpts is the number of kappa-points (chiral quantum number) we want \ to sample \ \>", "Subsubtitle"], Cell[BoxData[ \(\(Clear[SKnanotubebands]; \n SKnanotubebands[nn1_, nn2_, nkpts_] := Module[\n\t\t{othercoors, i, j, theta, rotxy, E1, E2, NC, H, B, BE, h, phi, Radius, aatom, batom, m1, m2, n1, n2, \n\t\t\tBtmp, BEtmp, S, invS, Cn, dbasis, hbasis, phibasis, \n\t\tbands, bandplots, eiglist, HAM, M, kappa, NS, dottmp, abhamlist}, \n\t\t\n\t\t n1 = Min[nn1, nn2]; \n\t\tn2 = Max[nn1, nn2]; \n\n\t\ \ Btmp = \((n1*e1 + n2*e2)\); \n\t\tBEtmp = Btmp/Sqrt[Btmp . Btmp]; \n \t\t\n\t\t (*\ redefine\ e1, \ e2\ so\ B, BE\ is\ along\ the\ x - axis, \ \(--\ the\)\ Tube\ is\ along\ y\ *) \n\t\ttheta = ArcCos[BEtmp . {1, 0, 0}]; \n\t\t rotxy = {{Cos[theta], Sin[theta], 0}, {\(-Sin[theta]\), Cos[theta], 0}, {0, 0, 1}}; \n\t\tE1 = rotxy . e1; \n\t\tE2 = rotxy . e2; \n \t\tB = {Sqrt[Btmp . Btmp], 0, 0}; \n\t\tBE = {1, 0, 0}; \n\t\t\n \t\t\tRadius = Sqrt[B . B]/\((2 Pi)\); \n\t\ \ Print["\", Radius]; \n\t\t\n \t\t (*\ Determine\ m1, m2\ *) \n\t\t\t\tNC = GCD[n1, n2]; \n \ \ \ \ \ \ \ \ \ Print["\", NC, "\<-order axis\>"]; \n \t\t\t\tm1 = 0.5; \n\t\tm2 = \(-1\); \ \n\t While[IntegerPart[m1] < m1, \n\t\t\tm2 = m2 + 1; \n\t\t\t m1 = \((1 + m2*\((n1/NC)\))\)/\((n2/NC)\); \n\t\t]; \n\t\t Print["\", m1, "\<,\>", m2]; \n\t\tH = E1*m1 + E2*m2; \n \t\th = Sqrt[\((Cross[H, B] . Cross[H, B])\)/\((B . B)\)]; \n\t\t phi = 2 Pi \((H . B)\)/\((B . B)\); \n\t\t Print["\<(h,phi) = (\>", h, "\<,\>", phi, "\<)\>"]; \n\n \t\t (*\ Chiral\ \(operator : \ translate\ h\ in\ y\ dir . \ and\ rotate\ phi\ around\ y\ axis \)\ *) \n\t\t\t\tClear[S]; \n\t S[r_] := \ \ {0, h, 0}\ + \((\ {{Cos[phi], 0, Sin[phi]}, {0, 1, 0}, {\(-Sin[phi]\), 0, Cos[phi]}} . r)\); \ \n\t\t\t\n\t\t\t\tClear[invS]; \n\t\t invS[r_] := \ \ {0, \(-h\), 0}\ + \((\ {{Cos[\(-phi\)], 0, Sin[\(-phi\)]}, {0, 1, 0}, { \(-Sin[\(-phi\)]\), 0, Cos[\(-phi\)]}} . r)\); \n\t\t\n \t\t (*\ Rotation\ operator\ Cn\ *) \n\t\tClear[Cn]; \n\t\t Cn[r_] := \ {{Cos[2 Pi/NC], 0, Sin[2 Pi/NC]}, {0, 1, 0}, { \(-Sin[2 Pi/NC]\), 0, Cos[2 Pi/NC]}} . r; \n\t\t\n \t\t (*\ basis\ \(atoms : \ Note\ the\ tube\ axis\ is\ the\ y - axis\)\ *) \n\t\n \t\taatom = {Radius, 0, 0}; \ \n\t\tdbasis = \((E1 + E2)\)/3\ ; \n\t\t hbasis = Sqrt[\((Cross[dbasis, {1, 0, 0}] . Cross[dbasis, {1, 0, 0}])\)]; \n \t\tphibasis = 2 Pi*\((dbasis . B)\)/\((B . B)\); \n\t\t batom = \t\ {0, hbasis, 0}\ + \n\t\t\t\t\t \((\ {{Cos[phibasis], 0, Sin[phibasis]}, {0, 1, 0}, { \(-Sin[phibasis]\), 0, Cos[phibasis]}} . {Radius, 0, 0}) \); \ \n\tPrint["\", phibasis]; \n\t\n \t\t (*\ nanotube\ y\ unit - cell\ *) \n\t\t dR = GCD[2*\((n1)\) + n2, 2*\((n2)\) + n1]; \n\t\t T = Sqrt[3]*2 Pi*Radius/dR; \n\t\t Print["\", T]; \t\n\t\t NT = T/h; \n\t\t\t Print["\", NT, \ "\< chiral operations\>"]; \t\n \t (*\ Generate\ bands\ from\ coordinates\ using\ symmetry\ operations \ *) \n\t\t\n \t\t (*\ Note\ we\ use\ that\ we\ have\ chiral\ "\"\ symmetry\ in\ the\ sense\ *) \n \t\t (*\ that\ \((1/S)\)\ generates\ an\ equivalent\ atom \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ *) \ \n\t\t \n\t\t (*\ We\ must\ find\ matrix\ elements\ which\ give\ non - zero\ value\ on\ the\ B\ basis\ \(state : \)\ *) \n \t\t (*\ Lets\ search\ among\ NS\ A - equivalent\ coordinates\ \((excl . \ A\ it\ self)\)\ *) \n\t\t NS = Sqrt[n1^2\ + \ n2^2] + 5; \n\n\t\t abhamlist = Flatten[\n \t\t\t\t{ Table[SKhamHOP12[aatom, Nest[Cn, Nest[invS, batom, i], j]], {j, 0, NC - 1}, {i, 1, NS}], \n\ \ \ \ \ \ \ \ \ \ Table[\ SKhamHOP12[aatom, Nest[Cn, Nest[\ \ \ \ \ \ \ S, batom, i], j]], {j, 0, NC - 1}, {i, 0, NS}]}, 2]; \n\n \t\t\t (*\ Bands\ using\ chiral\ kappa\ vector\ *) \t\n \t\t (*\ table\ of\ plots\ for\ different\ rotation\ symmetry\ *) \n \t\t\t\t\t bandplot = Table[Print[ "\< ======= Bands for rotational quantum number M = \>", M, "\< ======= \>"]; \ \n\t\t bands = Table[\n\t\t\t phaselist = Flatten[\n \t\t\t\t\t\t\t{\t Table[Exp[\(+Pi\)*I*kappa*i\ ]* Exp[\(-2\) Pi*I*M*j/NC]\ , {j, 0, NC - 1}, {i, 1, NS}], \n\t\t\t\t\t\t\t\t\ \ \ Table[Exp[\(-Pi\)*I*kappa*i\ ]* Exp[\(-2\) Pi*I*M*j/NC]\ , {j, 0, NC - 1}, {i, 0, NS}]}, 2]; \n\t\t\t\t\t\t\n\t\t\t\t\t\t dottmp = phaselist . abhamlist; \ \n\t\t\t\t\t\n \t\t\t\t\t\t\t\t\t\t HAM = \ dottmp\ + \ Conjugate[Transpose[dottmp]]\ + \ SKhamOnsite\ ; \t\t\n\t\t\n\t\t\t\n\t\t\t\t\t\t\t If[M == 0, \ ky = Mod[Pi*\((kappa)\)*T/h, 2 Pi], \n \t\t\t\t\t\t\t\ ky = Mod[ Pi*\((kappa)\)*T/h\ \ - \ \((2 Pi/NC)\)*M*\((\((phi*T/h)\)/\((2 Pi/NC)\))\), 2 Pi]]; \t\n\t\t\t\t\t\t\n\t\t\t\t\t\t eigtmp = \tEigenvalues[HAM]; \n\t\t\n\t\t\t\t\t\t\t Transpose[{Pi*kappa*{1, 1, 1, 1, 1, 1, 1, 1}, Chop[eigtmp]}], \n\t\t\t\t\t\t{kappa, \(-1\), 1, 1/nkpts}]; \ \n\n \t\t\t\t\t\t (* \t\t{\(Transpose[{\(-ky\)*{1, 1, 1, 1, 1, 1, 1, 1}, eigtmp}], \)\ *) \n \t\t\t\t\t\t (* \t\ \tTranspose[{\ ky*{1, 1, 1, 1, 1, 1, 1, 1}, eigtmp}]}\ *) \n \t\t\t (*\t ListPlot[Flatten[bands, 2], PlotStyle -> {AbsolutePointSize[3]}, \ *) \n \t\t\t\t (*\t\tPlotLabel -> M, Frame -> True, FrameLabel -> {\*"\"\<\!\(k\_y\)\>\"", "\<\[CurlyEpsilon]\>"}, \ *) \n\t\t\t\t\t (*\ PlotRange -> {{\(-Pi\), Pi}, Automatic}, \ *) \n\t\t\t\t (* \tFrameTicks -> {{\(-Pi\), \(-Pi\)/2, 0, Pi/2, Pi}, Automatic, {}, {}}]\t\ *) \n\t\t\t\t\n\t\t\t\t\t\t ListPlot[Flatten[bands, 1], PlotStyle -> {AbsolutePointSize[3]}, \ \n\t\t\t\t\t\tPlotLabel -> M, Frame -> True, FrameLabel -> {"\<\[Kappa]\>", "\<\[CurlyEpsilon]\>"}, PlotRange -> {{\(-Pi\), Pi}, Automatic}, \ \n\t\t\t\t\t FrameTicks -> {{\(-Pi\), \(-Pi\)/2, 0, Pi/2, Pi}, Automatic, {}, {}}]\t\n\t\t\t\t\t\t\t, \ \ {M, 0, NC - 1}]; \n\n\tReturn[bandplot]; \n]\n\t\t\t\t\n\t\)\)], "Input"], Cell[CellGroupData[{ Cell["Example: (5,5) Tube", "Section"], Cell[BoxData[ \(SKnanotubebands[5, 5, 30]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Example: (9,2) Tube", "Section"], Cell[BoxData[ \(SKnanotubebands[9, 2, 200]\)], "Input"] }, Closed]] }, Closed]] }, FrontEndVersion->"X 3.0", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{822, 644}, WindowMargins->{{156, Automatic}, {Automatic, 77}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False} ] (*********************************************************************** Cached data follows. 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