(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 4.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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The variable names should be consistent with the text description. \ Cells in ", StyleBox["Mathematica", FontSlant->"Italic"], " are defined by the little blue bars on the right. The black text below \ is in the first cell. Put your cursor in there somewhere and use Shift+Enter \ to execute the code in a cell. Then, execute subsequent cells. Sometimes \ they may take awhile to execute. You can see this when the blue bars on the \ right change to black during execution. The variables which should be \ changed depending on the material of the band calculation are given in red. \ Change the pseudopotentials until a few bandgaps in major crystal directions \ are consistent with measured values. At that point then the rest of the \ bands should be accurate. (* Comments appear in these enclosures *)" }], "Text"], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", StyleBox[\( (*\ Set\ the\ size\ of\ the\ matrix\ to\ solve\ at\ each\ k\ point . \ \ \ The\ matrix\ size\ is\ given\ by\ n, \ but\ set\ indirectly\ by\ using\ \(\(lim\)\(.\)\)\ *) \), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", RowBox[{ StyleBox["(*", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[\(lim\ of\ 5\ yields\ a\ 124 x124\ matrix\ with\ good\ accuracy . \ \ lower\ is\ \ \(\(faster\)\(.\)\)\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox["*)", FontColor->RGBColor[1, 0, 0]]}], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], RowBox[{ StyleBox[\(lim = 5;\), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(Clear[h, k, l, m, veck, Vs, Va, a, A]\), "\[IndentingNewLine]", \( (*\ Define\ basis\ reciprocal\ lattice\ vectors\ *) \), "\[IndentingNewLine]", \(vecb1 = {\(-1\), 1, 1};\), "\[IndentingNewLine]", \(vecb2 = {1, \(-1\), 1};\), "\[IndentingNewLine]", \(vecb3 = {1, 1, \(-1\)};\), "\[IndentingNewLine]", \( (*\ Offset\ because\ each\ cell\ has\ 2\ atoms . \ \ Use\ a\ \ symmetrical\ offset\ *) \), "\[IndentingNewLine]", \(vecT = {1\/8, 1\/8, 1\/8};\), "\[IndentingNewLine]", \(n = lim^3 - 1;\), "\[IndentingNewLine]", \( (*\ Functions\ used\ to\ help\ build\ matrix\ of\ reciprocal\ lattice\ \ vectors\ *) \), "\[IndentingNewLine]", \(h[m_] := Floor[\(m + n/2\)\/lim^2] - Floor[lim/2];\), "\[IndentingNewLine]", \(k[m_] := Floor[Mod[m + n/2, lim^2]\/lim] - Floor[lim/2];\), "\[IndentingNewLine]", \(l[m_] := Mod[m + n/2, lim] - Floor[lim/2];\), "\[IndentingNewLine]", \(K[m_] := h[m]*vecb1 + k[m]*vecb2 + l[m]*vecb3;\), "\[IndentingNewLine]", "\[IndentingNewLine]", \( (*\ Material\ parameters\ *) \), "\[IndentingNewLine]", StyleBox[\(a = 5.6533;\), FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox[\( (*\ lattice\ constant\ in\ Angstroms\ *) \), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(m0 = 9.11*10\^\(-31\);\), " ", \( (*\ electron\ mass\ in\ kg\ *) \), "\[IndentingNewLine]", \(hbarJs = 1.054*10\^\(-34\);\), " ", \( (*\ hbar\ in\ J - s\ *) \), "\[IndentingNewLine]", \(hbareV = 6.581*10\^\(-16\);\), " ", \( (*\ hbar\ in\ eV - s\ *) \), StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", StyleBox[\( (*\ Pseudopotentials\ in\ eV\ \(\((from\ Rydberg\ units, \ as\ is\ usual\ in\ publications)\)\ --\)\ Change\ the\ \ first\ number\ in\ each\ \(\(definition\)\(.\)\)\ *) \), FontColor->RGBColor[1, 0, 0]], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], \( (*\ The\ last\ number\ in\ each\ definition\ below\ identifies\ which\ \ form\ factor\ it\ is\ *) \), "\[IndentingNewLine]", \(Vs[m_] := 0\), StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], RowBox[{\(Vs[m_]\), ":=", RowBox[{ RowBox[{ StyleBox[\(-0.23\), FontColor->RGBColor[1, 0, 0]], "*", "13.6059"}], "/;", \(K[m] . K[m] \[Equal] 3\)}]}], "\[IndentingNewLine]", RowBox[{\(Vs[m_]\), ":=", RowBox[{ RowBox[{ StyleBox["0", FontColor->RGBColor[1, 0, 0]], "*", "13.6059"}], "/;", \(K[m] . K[m] \[Equal] 4\)}]}], "\[IndentingNewLine]", RowBox[{\(Vs[m_]\), ":=", RowBox[{ RowBox[{ StyleBox["0.01", FontColor->RGBColor[1, 0, 0]], "*", "13.6059"}], "/;", \(K[m] . K[m] \[Equal] 8\)}]}], "\[IndentingNewLine]", RowBox[{\(Vs[m_]\), ":=", RowBox[{ RowBox[{ StyleBox["0.06", FontColor->RGBColor[1, 0, 0]], "*", "13.6059"}], "/;", \(K[m] . K[m] \[Equal] 11\)}]}], "\[IndentingNewLine]", RowBox[{\(Va[m_]\), ":=", RowBox[{ RowBox[{ StyleBox["0.07", FontColor->RGBColor[1, 0, 0]], "*", "13.6059"}], "/;", \(K[m] . K[m] \[Equal] 3\)}]}], "\[IndentingNewLine]", RowBox[{\(Va[m_]\), ":=", RowBox[{ RowBox[{ StyleBox["0.05", FontColor->RGBColor[1, 0, 0]], "*", "13.6059"}], "/;", \(K[m] . K[m] \[Equal] 4\)}]}], "\[IndentingNewLine]", RowBox[{\(Va[m_]\), ":=", RowBox[{ RowBox[{ StyleBox["0", FontColor->RGBColor[1, 0, 0]], "*", "13.6059"}], "/;", \(K[m] . K[m] \[Equal] 8\)}]}], "\[IndentingNewLine]", RowBox[{\(Va[m_]\), ":=", RowBox[{ RowBox[{ StyleBox["0.01", FontColor->RGBColor[1, 0, 0]], "*", "13.6059"}], "/;", \(K[m] . K[m] \[Equal] 11\)}]}], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[1, 0, 0]], \(Va[m_] := 0\), "\[IndentingNewLine]", "\[IndentingNewLine]", \( (*\ Definition\ of\ the\ potential\ and\ matrix\ elements\ *) \), "\[IndentingNewLine]", \(V[m_] := Vs[m]*Cos[2\ \[Pi]\ K[m] . vecT] + \[ImaginaryI]*Va[m]* Sin[2\ \[Pi]\ K[m] . vecT];\), "\[IndentingNewLine]", \(A[i_, j_, veck_] := \(hbarJs*hbareV\)\/\(2\ m0\)* Abs[\((veck + K[i])\) . \((veck + K[i])\)]* KroneckerDelta[i, j]*\((\(2\ \[Pi]\)\/\(a*10\^\(-10\)\))\)^2 + V[i - j];\), "\[IndentingNewLine]", \( (*\ Output\ the\ matrix\ size\ n\ x\ \(n\ --\)\ This\ is\ n\ *) \), "\[IndentingNewLine]", "n"}]}]], "Input"], Cell[BoxData[ \(\(\( (*\ kgenerator\ generates\ k\ values\ for\ m\ = \ 0\ to\ 40\ to\ make\ a\ complete\ band\ diagram; \ m\ is\ the\ x\ axis\ of\ a\ band\ diagram\ *) \)\(\[IndentingNewLine]\)\ \( (*\ this\ is\ merely\ a\ function\ definition\ that\ is\ later\ used\ \ *) \)\(\[IndentingNewLine]\)\(kgenerator[ m_] := {0.5 - m/20, 0.5 - m/20, 0.5 - m/20} /; m \[LessEqual] 10\ (*\ L \(( .5, .5, .5)\)\ to\ \[CapitalGamma] \((0, 0, 0)\)\ 10\ points\ *) \[IndentingNewLine] kgenerator[m_] := {\((m - 10)\)/10, 0, 0} /; m > 10 && m \[LessEqual] 20\ (*\ \[CapitalGamma] \((0, 0, 0)\)\ to\ X \((1, 0, 0)\)\ 10\ points\ *) \[IndentingNewLine] kgenerator[m_] := {1, \((m - 20)\)/10, 0} /; m > 20 && m \[LessEqual] 25\ (*\ X \((1, 0, 0)\)\ to\ W \((1, .5, 0)\)\ 5\ points\ *) \[IndentingNewLine] kgenerator[m_] := {1 - \((m - 25)\)/20, 0.5 + \((m - 25)\)/20, 0} /; m > 25 && m \[LessEqual] 30\ (*\ W \((1, .5, 0)\)\ to\ K \(( .75, .75, 0)\)\ *) \[IndentingNewLine] \(\(kgenerator[m_]\)\(:=\)\({0.75 - \((m - 30)\)/\((40/3)\), 0.75 - \((m - 30)\)/\((40/3)\), 0}\)\(\ \)\( (*\ K\ to\ \[CapitalGamma]\ *) \)\)\)\)\)], "Input"], Cell[BoxData[ \(\(\( (*\ This\ solves\ the\ band\ diagram\ at\ three\ symmetry\ points . \ \ The\ \ list\ of\ values\ represent\ energies\ in\ eV\ of\ bands\ crossing\ the\ \ given\ symmetry\ point . \ \ They\ are\ not\ referenced\ to\ any\ energy . \ \ \ Only\ their\ relationship\ to\ each\ other\ matters . \ \ A\ full\ diagram\ \ calculation\ is\ in\ the\ next\ \(\(cell\)\(.\)\)\ \ *) \)\(\ \[IndentingNewLine]\)\( (*\ Solve\ the\ \[CapitalGamma]\ point\ \ *) \)\(\[IndentingNewLine]\)\(\(matrixA\ = \ Table[A[i, j, {0, 0, 0}], {i, \(-n\)/2, n/2}, {j, \(-n\)/2, n/2}];\)\[IndentingNewLine] gammalist = Take[Re[Eigenvalues[matrixA]], \(-10\)]\[IndentingNewLine] (*\ X\ Point\ *) \[IndentingNewLine] \(matrixA\ = \ Table[A[i, j, {1, 0, 0}], {i, \(-n\)/2, n/2}, {j, \(-n\)/2, n/2}];\)\[IndentingNewLine] xlist = Take[Re[Eigenvalues[matrixA]], \(-10\)]\[IndentingNewLine] (*\ L\ Point\ *) \[IndentingNewLine] \(matrixA\ = \ Table[A[i, j, { .5, .5, .5}], {i, \(-n\)/2, n/2}, {j, \(-n\)/2, n/2}];\)\[IndentingNewLine] llllist = Take[Re[Eigenvalues[matrixA]], \(-10\)]\)\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Output", " ", "the", " ", "entire", " ", "band", " ", "diagram", " ", "and", " ", "save", " ", "to", " ", "a", " ", RowBox[{"file", ".", " ", StyleBox["Enter", FontColor->RGBColor[1, 0, 0]]}], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox["the", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], StyleBox["filename", FontColor->RGBColor[1, 0, 0]], StyleBox[" ", FontColor->RGBColor[1, 0, 0]], RowBox[{ StyleBox["below", FontColor->RGBColor[1, 0, 0]], StyleBox[".", FontColor->RGBColor[1, 0, 0]], " ", "You"}], " ", "may", " ", "have", " ", "to", " ", "search", " ", "for", " ", "it"}], ",", " ", \(since\ Mathematica\ puts\ files\ in\ weird\ places . \ \ \ Open\ in\ Excel\ by\ importing\ as\ a\ comma\ separated\ text\ file . \ \ \ Each\ row\ represents\ a\ vertical\ slice\ of\ a\ band\ diagram . \ Top\ rows\ are\ left\ slices . \ \ The\ numbers\ are\ not\ \ referenced\ to\ any\ particular\ energy\), ",", " ", \(although\ the\ unit\ is\ correct\ in\ eV . \ \ Usually\ the\ \ values\ are\ shifted\ so\ that\ the\ highest\ valence\ band\ gets\ energy\ of\ \ zero . \ \ This\ takes\ some\ time\ to\ execute\)}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{\(outputband = Table[\[IndentingNewLine]matrixA = Table[A[i, j, kgenerator[m]], {i, \(-n\)/2, n/2}, {j, \(-n\)/2, n/2}]; \[IndentingNewLine]locallist = Take[Re[Eigenvalues[matrixA]] - 8.52357, \(-15\)]; Print[locallist]; locallist, {m, 0, 40}]\), "\[IndentingNewLine]", StyleBox[\( (*\ filename\ in\ quotes\ next\ line\ *) \), FontColor->RGBColor[1, 0, 0]], "\[IndentingNewLine]", \(Export["\", outputband, "\"];\)}]}]], "Input"] }, Open ]] }, FrontEndVersion->"4.2 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowSize->{1016, 668}, WindowMargins->{{0, Automatic}, {Automatic, 0}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 40, 0, 115, "Title"], Cell[1819, 55, 1176, 20, 147, "Text"], Cell[2998, 77, 6252, 136, 907, "Input"], Cell[9253, 215, 1285, 23, 150, "Input"], Cell[10541, 240, 1213, 23, 250, "Input"], Cell[11757, 265, 2197, 45, 210, "Input"] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)