(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' 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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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1276547, 64585]*) (*NotebookOutlinePosition[ 1277967, 64629]*) (* CellTagsIndexPosition[ 1277783, 64621]*) (*WindowFrame->Normal*) Notebook[{ Cell["Square Well Single Electron quantum Dot (SWSED)", "Section"], Cell[TextData[{ StyleBox["The package contained in the file \"SWSED.m\" calculates to high \ accuracy the conduction band structure for a single-electron in a \ three-dimensional spherical quantum dot within the model of finite \ square-well potential of radius ", FontFamily->"Times New Roman"], StyleBox["R", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" and energy barrier ", FontFamily->"Times New Roman"], StyleBox["V", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[". 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"DisplayMath", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[\(the\ energy\ barier\ of\ the\ finite\ square\ well . \ User\ input\ V\ in\ unit\ of\ Electron\ Volt\), "DisplayMath", FontFamily->"Times New Roman"]}, { RowBox[{ StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", "DisplayMath", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["Inside", "DisplayMath", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"]}], StyleBox[\(effective\ mass\ of\ \ electron\ inside\ the\ SWSED\), "DisplayMath", FontFamily->"Times New Roman"]}, { RowBox[{ StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", "DisplayMath", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["Outside", "DisplayMath", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"]}], StyleBox[\(effective\ mass\ of\ \ electron\ outside\ the\ SWSED\), "DisplayMath", FontFamily->"Times New Roman"]}, { RowBox[{ StyleBox["\[Epsilon]", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", "DisplayMath", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["Inside", "DisplayMath", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"]}], StyleBox[\(electric\ permeability\ inside\ the\ SWSED\), "DisplayMath", FontFamily->"Times New Roman"]}, { RowBox[{ StyleBox["\[Epsilon]", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", "DisplayMath", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["Outside", "DisplayMath", FontFamily->"Times New Roman", FontWeight->"Bold", FontSlant->"Italic"]}], StyleBox[\(electric\ permeability\ outside\ the\ SWSED\), "DisplayMath", FontFamily->"Times New Roman"]} }]], "DefinitionBox"], Cell[TextData[{ StyleBox["The effective mass approximation is proven to be justified for \ the determination of the first levels even for dot radius as small as 15 \ \[CapitalARing] (see reference [2], in the end of this manual). The package \ is developped by ", FontFamily->"Times New Roman"], "Tran The Trung and Nguyen Hong Quang at Institute of Physics, 46 Nguyen \ Van Ngoc, Cong Vi, Ba \[CapitalEth]\[IGrave]nh, H\[AGrave] Noi, Vietnam in \ 2003." }], "Text"], Cell["Energy Level", "Section"], Cell[TextData[{ StyleBox["The main function of the package is EnergyLevel. We can get \ information about this function by the standard ", FontFamily->"Times New Roman"], StyleBox["Mathematica", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" command ", FontFamily->"Times New Roman"], StyleBox["?", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[" placed before the name of the function.", FontFamily->"Times New Roman"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?EnergyLevel\)\)], "Input", CellLabel->"In[3]:="], Cell[BoxData[ \("EnergyLevel[{R,V},{l,n}] gives energy in eV of state of the single \ electron indicated by l and n within the square well quantum dot specified by \ {R,V} with default material as Si inside and SiO2 outside. Other material \ can be specified by using options EMass and Permeability. Type ?EMass and \ ?Permeability for more help on these options."\)], "Print", CellTags->"Info3300515126-3940013"] }, Open ]], Cell[BoxData[GridBox[{ {\(EnergyLevel[{R, V}, {l, n}]\), \(gives\ energy\ in\ eV\ of\ state\ of\ the\ single\ \ electron\ indicated\ by\ l\ and\ n\ within\ the\ square\ well\ quantum\ dot\ \ specified\ by\ \(\({R, V, Min, Mout}\)\(.\)\)\)}, {"EMass", \(option\ to\ specify\ effective\ electron\ mass\ inside\ \ and\ out\ side\ the\ SWSED . \ Default\ value\ is\ {0.27, \ 0.5}\ meaning\ electron\ mass\ inside\ the\ dot\ is\ 0.27\ \ and\ out\ side\ the\ dot\ is\ 0.5 . \ These\ values\ are\ correct\ for\ a\ Si - SiO2\ SWSED . \ Other\ material\ may\ require\ other\ \(\(values\)\(.\)\)\)}, {"Permeability", \(option\ to\ specify\ electric\ permeability\ \ inside\ and\ out\ side\ the\ SWSED . \ Default\ value\ is\ {11.7, \ 11.7}\ meaning\ permeability\ inside\ the\ dot\ is\ 11.7\ and\ \ out\ side\ the\ dot\ is\ 11.7 . \ These\ values\ are\ correct\ for\ a\ Si - SiO2\ SWSED . \ Other\ material\ may\ require\ other\ \(\(values\)\(.\)\)\)} }]], "DefinitionBox"], Cell[TextData[{ StyleBox["For input to EnergyLevel, we have 2 list. The first list is the 2 \ real positive numbers that specifies the SWSED.", FontFamily->"Times New Roman"], " ", StyleBox["The second list contains 2 integer ", FontFamily->"Times New Roman"], StyleBox["l", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" and ", FontFamily->"Times New Roman"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[", being the quantum angular state and quantum energy state \ respectively. Note that ", FontFamily->"Times New Roman"], StyleBox["l", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" must be equal or greater than 0 and ", FontFamily->"Times New Roman"], StyleBox["n", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" must be equal or greater than 1. 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For example: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EnergyLevel[{20, 10000000}, {1, 2}]\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(EnergyLevel::"bessel" \(\(:\)\(\ \)\) "Matching equation badly behaves near root, EnergyLevel assumes root \ close to Bessel root."\)], "Message", CellLabel->"From In[12]:="], Cell[BoxData[ \(2.1044681851793605`\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell["\<\ One get the same result by setting V = \[Infinity]: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EnergyLevel[{20, \[Infinity]}, {1, 2}]\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \(2.1044681851793605`\)], "Output", CellLabel->"Out[14]="] }, Open ]], Cell["Energy Structure", "Section"], Cell[TextData[{ "Bound state of the single electron in SWSED is the state that has energy \ level lower than barrier. If barrier is sufficiently low while quantum energy \ state number ", StyleBox["n", FontSlant->"Italic"], " is high, the energy level may exceeds barrier and the state is unstable:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EnergyLevel[{20, 1.2}, {1, 2}]\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \(EnergyLevel::"unbound" \(\(:\)\(\ \)\) "EnergyLevel encounters level demanded unbound or potential well too \ shallow."\)], "Message", CellLabel->"From In[16]:="], Cell[BoxData[ \(EnergyLevel::"noroot" \(\(:\)\(\ \)\) "EnergyLevel finds no bound energy level, output Null."\)], "Message", CellLabel->"From In[16]:="] }, Open ]], Cell[TextData[{ "One can find all bound states of the single electron for a given SWSED by \ trying all different pairs of quantum number ", StyleBox["l", FontSlant->"Italic"], " and ", StyleBox["n", FontSlant->"Italic"], " which gives bound energy levels. However, the SWSED Package provides a \ built-in function that give exactly this, the function EnergyStructure." }], "Text"], Cell[BoxData[GridBox[{ {\(EnergyStructure[R, V]\), \(gives\ energies\ \((in\ eV)\)\ of\ all\ bound\ states\ of\ \ the\ single\ electron\ within\ the\ square\ well\ quantum\ dot\ specified\ \ by\ {R, V, Min, Mout}\)}, {"EMass", \(option\ to\ specify\ effective\ electron\ mass\ inside\ \ and\ out\ side\ the\ SWSED . \ Default\ value\ is\ {0.27, \ 0.5}\ meaning\ electron\ mass\ inside\ the\ dot\ is\ 0.27\ \ and\ out\ side\ the\ dot\ is\ 0.5 . \ These\ values\ are\ correct\ for\ a\ Si - SiO2\ SWSED . \ Other\ material\ may\ require\ other\ \(\(values\)\(.\)\)\)}, {"Permeability", \(option\ to\ specify\ electric\ permeability\ \ inside\ and\ out\ side\ the\ SWSED . \ Default\ value\ is\ {11.7, \ 11.7}\ meaning\ permeability\ inside\ the\ dot\ is\ 11.7\ and\ \ out\ side\ the\ dot\ is\ 11.7 . \ These\ values\ are\ correct\ for\ a\ Si - SiO2\ SWSED . \ Other\ material\ may\ require\ other\ \(\(values\)\(.\)\)\)} }]], "DefinitionBox"], Cell[TextData[{ "The output of EnergyStructure is a list of 2 elements. The first element \ is the list of 6 real positive numbers that specifies the SWSED (", StyleBox["R", FontWeight->"Bold", FontSlant->"Italic"], ", ", StyleBox["V", FontWeight->"Bold", FontSlant->"Italic"], ", ", StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" inside", FontWeight->"Bold", FontSlant->"Italic"], ", ", StyleBox["m", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" outside, \[Epsilon] inside, \[Epsilon] outside", FontWeight->"Bold", FontSlant->"Italic"], "). The second element has depth of 2. The first depth of this list \ correponds to quantum angular number ", StyleBox["l", FontSlant->"Italic"], " = 0, 1, 2.... The second depth are list of energy levels that has the \ same quantum angular number ", StyleBox["l", FontSlant->"Italic"], " but different quantum state number ", StyleBox["n", FontSlant->"Italic"], " = 0, 1, ... Options of EnergyStructure also include EMass and \ Permeability as for EnergyLevel. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EnergyStructure[20, 3.1]\)], "Input", CellLabel->"In[18]:="], Cell[BoxData[ \({{20, 3.1`, 0.27`, 0.5`, 11.7`, 11.7`}, {{0.26158937493995515`, 1.0538472404421722`, 2.3670879522238675`}, {0.5357196582679422`, 1.5936372647685968`, 3.0837789727234632`}, {0.880836223939068`, 2.2004407252835585`}, {1.2917265237737074`, 2.8532228562552757`}, {1.7630040750121896`}, {2.2878216790554466`}, \ {2.8555509285981593`}}}\)], "Output", CellLabel->"Out[18]="] }, Open ]], Cell["\<\ It is much easier to view the whole energy structure graphically, by using the function StructurePlot of SWSED package. \ \>", "Text"], Cell[BoxData[GridBox[{ { RowBox[{"StructurePlot", "[", RowBox[{ StyleBox["Output", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox["of", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], "EnergyStructure"}], "]"}], \(plots\ all\ bound\ states\ calculated\ by\ \ EnergyStructure\ of\ the\ single\ electron\ within\ 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PlotPDF is False. The option PlotPDF will be discussed in the next section. From the graphics of the energy structure, one can find the order of energy level, for example above, as 1s < 2p <3d <2s < 4f < 3d ... The package SWSED comes with a function that help automatically give the order of the energy level structure: StructureOrder. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(StructureOrder[EnergyStructure[20, 3.1]]\)], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \({{20, 3.1`, 0.27`, 0.5`, 11.7`, 11.7`}, {"1s", "2p", "3d", "2s", "4f", "3p", "5g", "4d", "6h", "3s", "5f", "7i", "4p"}}\)], "Output", CellLabel->"Out[28]="] }, Open ]], Cell["Note that StructureOrder has no option:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(StructureOrder[EnergyStructure[30, 3.1\ ], EMass \[Rule] {0.27, 0.27}]\)], "Input", CellLabel->"In[30]:="], Cell[BoxData[ \(StructureOrder::"noop" \(\(:\)\(\ \)\) "StructureOrder has no option. Type ?StructureOrder for the usage of \ this function. Also type ? infront of the option name for the usage of the \ option you intend to use."\)], "Message", CellLabel->"From In[30]:="] }, Open ]], Cell["The option EMass is reserved to EnergyStructure function:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?EMass\)\)], "Input", CellLabel->"In[32]:="], Cell[BoxData[ \("EMass is an option for EnergyLevel, EnergyStructure, \ RadialWaveFunction and DensityFunction functions, it can be specified by a \ list of two positive numbers, representing electron effective mass inside and \ outside of SWSED, in unit of free electron mass. Default value is {0.27, \ 0.5}, which is correct for Si-SiO2 dot."\)], "Print", CellTags->"Info3300515135-8358773"] }, Open ]], Cell["\<\ So the energy order for SWSED with R = 30, V = 3.1eV and electron mass both 0.27 inside and outside can be found by: \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(StructureOrder[ EnergyStructure[30, 3.1\ , EMass \[Rule] {0.27, 0.27}]]\)], "Input", CellLabel->"In[34]:="], Cell[BoxData[ \({{30, 3.1`, 0.27`, 0.27`, 11.7`, 11.7`}, {"1s", "2p", "3d", "2s", "4f", "3p", "5g", "4d", "6h", "3s", "5f", "7i", "4p", "8k", "6g", "5d", "4s", "9l", "7h", "6f", "10m", "5p", "8i", "7g", "11n"}}\)], "Output",\ CellLabel->"Out[34]="] }, Open ]], Cell["\<\ As input to StructurePlot and StructureOrder are output of EnergyStructure, we also provide a function EnergyStructureQ, which tests if a data can be possibly an output of EnergyStructure \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?EnergyStructureQ\)\)], "Input", CellLabel->"In[36]:="], Cell[BoxData[ \("EnergyStructure[data] gives True if data is output of EnergyStructure \ function, and False ortherwise. EnergyStructureQ has no option."\)], "Print", CellTags->"Info3300515136-9825092"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(EnergyStructureQ[ EnergyStructure[30, 3.1\ , EMass \[Rule] {0.27, 0.27}]]\)], "Input", CellLabel->"In[37]:="], Cell[BoxData[ \(True\)], "Output", CellLabel->"Out[37]="] }, Open ]], Cell["\<\ You can use this function in case you are not sure if input to StructurePlot or StructureOrder are corrected. \ \>", "Text"], Cell["Wave Function & Probability Density Function", "Section"], Cell[TextData[{ "Apart from energy level information, one can also find the wave function \ of the single electron in a given SWSED at a bound state. This wave function \ is a product of spherical harmonic ", Cell[BoxData[ \(TraditionalForm\`Y\_\(l, m\)\)]], ", with orbital quantum number ", StyleBox["l", FontSlant->"Italic"], " (", StyleBox["l", FontSlant->"Italic"], " = 0, 1, 2, \[Ellipsis]) and azimuthal quantum number ", StyleBox["m", FontSlant->"Italic"], " (-", StyleBox["l", FontSlant->"Italic"], " \[LessEqual] ", StyleBox["m", FontSlant->"Italic"], " \[LessEqual] ", StyleBox["l", FontSlant->"Italic"], "), and radial part F(r).\n\t\[CapitalPsi](", Cell[BoxData[ \(TraditionalForm\`\(r\& \[Rule] \)\)]], ") = ", Cell[BoxData[ \(TraditionalForm\`Y\_\(l, m\)\)]], "(\[CurlyPhi], \[Theta]) F(r)\t\t\nF(r) depend on the SWSED parameters and \ the state, which can be calculated by the function RadialWaveFunction. See \ the appended article in the end of this help file for the analytic form of \ F(r)." }], "Text"], Cell[BoxData[GridBox[{ {\(RadialWaveFunction[{R, V}, {l, n}]\), \(gives\ radial\ wave\ function\ in\ standard\ \ Mathematica\ function\ format\ of\ the\ single\ electron\ inside\ square\ \ well\ quantum\ dot\ {R, V, Min, Mout}\ at\ the\ state\ {l, n}\)} }]], "DefinitionBox"], Cell[TextData[{ "RadialWaveFunction also has the options EMass and Permeability for \ specification of material of the SWSED other than Si-SiO2. 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states in Si-nanocrystal \ embedded in SiO", FontFamily->"Times New Roman"], StyleBox["2", FontFamily->"Times New Roman", FontSize->9], StyleBox["\"", FontFamily->"Times New Roman"], " below\n[2] J. S\[EAcute]e, P. Dollfus, and S. Galdin, Phys. Rev. B 66 \ (2002) 193307\n[3] S. Wolfram, The Mathematica Book, Cambridge University \ Press, (1999)" }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Exact calculation of single-electron states in Si-nanocrystal \ embedded in SiO", FontFamily->"Arial"], StyleBox["2", FontFamily->"Arial", FontSize->9] }], "SubsectionIcon"], Cell["\<\ Authors: Nguyen Hong Quang (a,b), Tran The Trung (a,c), Johann S\[EAcute]e (b) and Philippe Dollfus (b), \ \>", "Text"], Cell["\<\ a Institute of Physics,NCST,46 Nguyen Van Ngoc,Hanoi,Vietnam b Institut d\[CloseCurlyQuote]Electronique Fondamentale,B\[AHat]timent 220,UMR 8622-CNRS-Universit\[EAcute] Paris-Sud XI,91405 Orsay,France c Service d'A\[EAcute]ronomie,R\[EAcute]duit de Verri\[EGrave]res,B.P.3,Route \ des G\[AHat]tines,91371 Verri\[EGrave]res le Buisson,France \ \>", "Commentary"], Cell[TextData[{ "Abstract\nWe present an exact calculation of the single-electron energies \ and wave-functions for any bound state in a realistic Si-SiO2spherical \ quantum dot,including the material dependence of the electron effective mass. \ The influence of dot radius,confinement barrier potential and barrier-to-dot \ electron mass ratio on the electronic structure is investigated in detail. \ The results show that the energy structure shifts down from some tens to some \ hundreds meV compared to that obtained in the simplified model where the \ change in effective mass is neglected. Our exact single-electron calculation \ is finally used to verify the accuracy of the results obtained from a \ numerical approximate method developed to treat many-electron systems.\nPACS \ numbers:71.15.La,73.20.DxKeywords:A.Semiconductors; A.Nanostructures; \ D.Electronic states\n", StyleBox["Corresponding author. \tE-mail\[NonBreakingSpace]: \ philippe.dollfus@ief.u-psud.fr\n\t\t\t\tFax\[NonBreakingSpace]: +33 (0)1 69 \ 15 40 20", FontSize->10] }], "Text"], Cell["1. Introduction", "Subsection"], Cell["\<\ Efforts to integrate sub-10 nm-scaled CMOS devices will face physical limitations in the near future. To continue further increases in density with \ low power consumption, the trend is now to develop new device concepts turning the size reduction and quantum effects to advantage. In this field, one of the most promising idea is to use the Coulomb blockade mechanism in conducting island/tunnel junction systems to accurately control the current in single electron devices (SEDs).Self-assembled silicon nanocrystals (NCs) embedded in SiO2 appear as a good basic structure for the realization of SEDs \ likely to operate at room temperature[1]. The compatibility of such NCs with Si technology makes them good candidates for actual applications in combination with conventional CMOS circuits. The concept of multi-dot memory using Si NCs as a floating gate has been demonstrated[2-4]. It could be developed to design nonvolatile multibit memory cells of very high scalability and reliability with low power consumption. The Coulomb blockade and quantization effects have been used in memory device with self-aligned doubly-stacked dots to improve the retention time[5] and demonstrated in single electron transistors (SETs) at room temperature[6]. The realisation and optimisation of SEDs requires accurate computational simulation of single electron charging effects to improve our understanding of the underlying physics. One important preliminary task is to calculate exactly the energy band structure inside the semiconductor NC where quantum effects become important. Several numerical techniques have been developed using either a tight-binding description[7] or an effective mass approximation[8-10]. The self-consistent Poisson-Schr\[ODoubleDot]dinger calculation based on the Hartree approximation or the density functional theory is convenient to determine the ground state of a many-electron system in the dot as in Ref.10. Simple analytical models have been reported in the case of a single electron in a spherical quantum dot[11,12]. However,in these \ works the authors assumed the electron effective mass to be the same in the dot as in the surrounding barrier,which can make the results inappropriate to \ realistic applications. The present work deals with the exact calculation of single-electron energy and wave-function for all bound states in a realistic Si NC embedded in SiO2by including the change in effective mass from one material to the other. We give a detailed investigation of the energy structure for a wide range of dot radius,barrier height,and barrier-to-dot electron mass ratio. Our results are finally used to verify the accuracy of the results obtained using the numerical method developed in Ref.10. It should be noted that the calculation is exact, fast and applicable for a spherical square-well quantum dot system of any size and potential barrier in \ any material system. \ \>", "Text"], Cell["2. Model for Exact Calculation", "Subsection"], Cell[TextData[{ "We study the conduction band structure for a single-electron in a \ spherical QD within the model of finite square-well potential of radiusRand \ energy barrier. Unless otherwise indicated, the effective atomic units are \ used in the whole article, i.e.the unit of mass is the free electron mass m", StyleBox["0", FontSize->9], ", the unit of length is the effective Bohr radius, ", Cell[BoxData[ FormBox[ RowBox[{\(a\_B\^*\), "=", RowBox[{ SubscriptBox["a", StyleBox["B", FontWeight->"Bold"]], "/", \(m\_Si\)}]}], TraditionalForm]]], " (with ", Cell[BoxData[ FormBox[ SubscriptBox["a", StyleBox["B", FontWeight->"Bold"]], TraditionalForm]]], " = 0.529 \[CapitalARing]) and the unit of energy is the effective Hartree \ ", Cell[BoxData[ \(TraditionalForm\`\(E\_Hart\^*\) = m\_Si\ E\_Hart\)]], " (with ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(E\_Hart\)\)\)]], " = 27.212 eV).The electron effective mass in silicon ", Cell[BoxData[ \(TraditionalForm\`m\_Si\)]], " is approximated as the harmonic mean of the longitudinal and transverse \ masses in \[CapitalDelta] valleys, i.e. ", Cell[BoxData[ \(TraditionalForm\`m\_Si = 3 \((1/m\_l\ + \ 2/m\_t)\)\^\(-1\)\)]], "= 0.27. On the basis of Weinberg experimental work [13] the barrier height \ ", Cell[BoxData[ \(TraditionalForm\`V\_0\)]], " and the electron effective mass in SiO2 ", Cell[BoxData[ \(TraditionalForm\`m\_Si02\)]], " are chosen equal to 3.1 eV and 0.5, respectively. These values are \ frequently used in tunnelling calculations in the Si/SiO2 system (see[14] and \ references therein). \nIn the effective mass approximation, which proves to \ be justified for the determination of the first levels even for dot radius as \ small as 15 \[CapitalARing][15], the single electron Schr\[ODoubleDot]dinger \ equation is given by\n\t[-1/2 ", Cell[BoxData[ FormBox[ RowBox[{ FormBox[\(\[Del]\_\(r\& \[Rule] \)\), "TraditionalForm"], \((\(1\/\(m\^*\)\) \[Del]\_\(r\& \[Rule] \ \))\)}], TraditionalForm]]], " + V(", Cell[BoxData[ \(TraditionalForm\`\(r\& \[Rule] \)\)]], ") ]\[CapitalPsi](", Cell[BoxData[ \(TraditionalForm\`\(r\& \[Rule] \)\)]], ") = E \[CapitalPsi](", Cell[BoxData[ \(TraditionalForm\`\(r\& \[Rule] \)\)]], ")\t (1)\nwhere\n\tV(", Cell[BoxData[ \(TraditionalForm\`\(r\& \[Rule] \)\)]], ") = ", Cell[BoxData[ FormBox[ RowBox[{"{", GridBox[{ {\(0\ , \ r < R\)}, {\(V\_0\ , \ r \[GreaterEqual] \ R\)} }]}], TraditionalForm]]], "\t\tand \t\tm* = ", Cell[BoxData[ RowBox[{"{", GridBox[{ {\(m\_\(Si\ , \ \ r\ < \ R\)\)}, {\(m\_SiO2\ , \ r \[GreaterEqual] R\)} }]}]]], "\t\t(2)\nDue to spherical symmetry of the system, the wavefunction may be \ written in the form\n\t\[CapitalPsi](", Cell[BoxData[ \(TraditionalForm\`\(r\& \[Rule] \)\)]], ") = ", Cell[BoxData[ \(TraditionalForm\`Y\_\(l, m\)\)]], "(\[CurlyPhi], \[Theta]) F(r)\t\t(3)\nwhere ", Cell[BoxData[ \(TraditionalForm\`Y\_\(l, m\)\)]], " is a spherical harmonic with orbital quantum number ", StyleBox["l", FontSlant->"Italic"], " (", StyleBox["l", FontSlant->"Italic"], " = 0, 1, 2, \[Ellipsis]) and azimuthal quantum number ", StyleBox["m", FontSlant->"Italic"], " (-", StyleBox["l", FontSlant->"Italic"], " \[LessEqual] ", StyleBox["m", FontSlant->"Italic"], " \[LessEqual] ", StyleBox["l", FontSlant->"Italic"], "). It can be easily shown from Eq. (1) and (2) that the differential \ equations for the radial part F(r) inside and outside the dot are just the \ equations of spherical ordinary and modified Bessel functions, ", Cell[BoxData[ \(TraditionalForm\`j\_l\)]], "(r) and ", Cell[BoxData[ \(TraditionalForm\`k\_l\)]], "(r), respectively. The solutions, which are finite at the center of the \ dot (r\[NonBreakingSpace]=0) and tend to zero whenrincreases indefinitely are \ given by\n-inside the dot:\t\n\tF(r) = A ", Cell[BoxData[ \(TraditionalForm\`j\_l\)]], "(\[Alpha] r) = A ", Cell[BoxData[ \(TraditionalForm\`\(\@\(\[Pi]\/\(2 \[Alpha]r\)\)\) \(\(J\_\(l + 1/2\)\)(\[Alpha]r)\)\)]], "; with \[Alpha] = ", Cell[BoxData[ \(TraditionalForm\`\@E\)]], "\t\t(4)\n-outside the dot:\n\tF(r) = B ", Cell[BoxData[ \(TraditionalForm\`k\_l\)]], "(\[Beta]r)\t= B", Cell[BoxData[ \(TraditionalForm\`\(\@\(\[Pi]\/\(2 \[Beta]r\)\)\) \(\(K\_\(l + 1/2\)\)(\[Beta]r)\)\)]], " ; with \[Beta] = ", Cell[BoxData[ \(TraditionalForm\`\@\(\[Gamma](V\_0 - E)\)\)]], ", \[Gamma] = ", Cell[BoxData[ \(TraditionalForm\`m\_SiO2\/m\_Si\)]], "\t(5)\nwhere ", Cell[BoxData[ \(TraditionalForm\`J\_\(l + 1/2\)\)]], "(\[Alpha]r) is the Bessel function of the first kind and ", Cell[BoxData[ \(TraditionalForm\`K\_\(l + 1/2\)\)]], "(\[Beta]r) is the modified Bessel function of the second kind. The \ normalization coefficient B is defined from the continuity condition imposed \ on the wavefunction at the dot surface: B = A ", Cell[BoxData[ \(TraditionalForm\`j\_l\)]], "(\[Alpha]R)/", Cell[BoxData[ \(TraditionalForm\`k\_l\)]], "(\[Beta]R). From the normalization of the wavefunction, A is given by\n\tA \ = ", Cell[BoxData[ \(TraditionalForm\`\@\(T1 + T2\)\)]], ", where T1 = ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_0\%R\(\(|\)\(\(j\_l\)(\[Alpha]r)\)\( | \ \^2\)\(\(r\^2\) \[DifferentialD]r\)\)\)]], ", T2 = ", Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(\(\(j\_l\)(\[Alpha]R)\)\/\(\(k\_l\)(\[Beta] R)\)\)\( | \^2\)\(\[Integral]\_R\%\[Infinity]\(\(|\)\(\(k\_l\ \)(\[Beta]r)\)\( | \^2\)\(\(r\^2\) \[DifferentialD]r\)\)\)\)\)]], "\t(6)\nTo determine the energy levels in the dot, we use the \ transcendental equation resulting from the boundary condition imposed on the \ derivative of the wave-function at the dot surface, i.e., by taking into \ account the change in effective mass between Si and SiO2,\n\t", Cell[BoxData[ \(TraditionalForm\`j\_l\)]], "'(\[Alpha]R) = ", Cell[BoxData[ \(TraditionalForm\`\(1\/\[Gamma]\) \(\(j\_l\)(\[Alpha]R)\)\/\(\(k\_l\)(\ \[Beta]R)\)\)]], Cell[BoxData[ \(TraditionalForm\`k\_l\)]], "'(\[Beta]R)\t(7)\nUsing the following formulas for derivatives\n\t ", Cell[BoxData[ \(TraditionalForm\`j\_l\)]], "'(r) = ", Cell[BoxData[ \(TraditionalForm\`j\_\(l - 1\)\)]], "(r) - ", Cell[BoxData[ \(TraditionalForm\`\(l + 1\)\/r\)]], Cell[BoxData[ \(TraditionalForm\`j\_l\)]], "(r); and ", Cell[BoxData[ \(TraditionalForm\`k\_l\)]], "'(r) = -", Cell[BoxData[ \(TraditionalForm\`k\_\(l - 1\)\)]], "(r) - ", Cell[BoxData[ \(TraditionalForm\`\(l + 1\)\/R\)]], Cell[BoxData[ \(TraditionalForm\`k\_l\)]], "(r)\t\t(8)\nwe get the following equation from which the discrete values \ of electron energy E = ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\^2\)]], " = ", Cell[BoxData[ \(TraditionalForm\`V\_0\)]], " - ", Cell[BoxData[ \(TraditionalForm\`\[Beta]\^2\)]], "/\[Gamma] can be obtained:\n\t\[Gamma]", Cell[BoxData[ \(TraditionalForm\`\([\(\[Alpha]\ \(\(j\_\(l - 1\)\)(\[Alpha]R)\)\)\/\(\ \(j\_l\)(\[Alpha]R)\) - \(l + 1\)\/R]\)\ = \ \(-\(\(\[Beta]\ \(\(k\_\(l - 1\)\)(\(\@\(\[Gamma](V\_0 - \[Alpha]\^2)\)\) R)\)\)\/\(\(k\_l\)(\(\@\(\[Gamma]( V\_0 - \[Alpha]\^2)\)\) R)\)\)\) - \(l + 1\)\/R\)]], "\t\t(9)\nNote that, if we ignore the difference of electron effective mass \ between the different materials, the equation (9) simplifies to\n\t ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Alpha]j\_\(l - 1\)\)(\[Alpha]R)\)\/\(\(j\_l\)(\ \[Alpha]R)\)\)]], "=", Cell[BoxData[ \(TraditionalForm\`\(\(\[Beta]k\_\(l - 1\)\)(\[Beta]R)\)\/\(\(k\_l\)(\ \[Beta]R)\)\)]], ", with ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\^2\ + \ \[Beta]\^2\)]], "= ", Cell[BoxData[ \(TraditionalForm\`V\_0\)]], "\t\t(10)\nwhich has been used by various authors [11-12]. \nOnce the \ energies ", Cell[BoxData[ \(TraditionalForm\`E\_\(l, n\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_\(l, n\)\^2\)]], " which satisfy Eq. (9) are determined, the exact wave-functions of bound \ states are given by\t", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalPsi]\_\(l, m, n\)\)(\(r\& \[Rule] \))\)]], " = ", Cell[BoxData[ RowBox[{"{", "\t", GridBox[{ {\(\(Aj\_l\) \((\(\[Alpha]\_\(l, n\)\) r)\) \(Y\_\(l, m\)\) \((\[CurlyPhi], \[Theta])\)\ , \ r < R\)}, {\(A \(\(\( j\_l\) \((\(\[Alpha]\_\(l, n\)\) R)\)\)\/\(\(k\_l\) \((\(\@\(\[Gamma] \((V\_0 - \ \[Alpha]\_\(l, n\)\^2)\)\)\) R)\)\)\) \(k\_l\) \((\(\@\(\[Gamma] \((V\_0 - \ \[Alpha]\_\(l, n\)\^2)\)\)\) r)\) \(Y\_\(l, m\)\) \((\[CurlyPhi], \[Theta])\)\ , \ \ r \ \[GreaterEqual] R\)} }]}]]], " \t(11)\nwith A defined by Eq.(6). Given the radius R, the confinement \ barrier height ", Cell[BoxData[ \(TraditionalForm\`V\_0\)]], " and the mass ratio \[Gamma], Eq.(9) is solved to determine all allowed \ bound statesin the dot. The solution has been carried out by computation code \ written in ", StyleBox["Mathematica", FontSlant->"Italic"], " environment [16], yielding high accuracy within manageable calculation \ time." }], "Text"], Cell["3. Result and Discussion", "Subsection"], Cell["\<\ We are interested in spherical Si-SiO2 quantum dots with typical radius R in the range 10\[CapitalARing]-60\[CapitalARing]. The barrier energy V0 is 3.1 eV, and the mass ratio is \[Gamma]\[NonBreakingSpace]=1.85. However, for purposes of comparison and to analyse the influence of these parameters, V0 and \[Gamma] have been considered to vary in the range 1 eV-30 eV and 0.1 - 2.5, respectively. We plot in Fig.1 the R dependence of the 1s-level for \[Gamma]\[NonBreakingSpace]=1 and for V0\[NonBreakingSpace]= \[NonBreakingSpace]1 eV, 3.1 eV and 30 eV. \ \>", "Text"], Cell[CellGroupData[{ Cell["graphics 1", "SubsubsectionIcon"], Cell[BoxData[ \(\(L = {\[IndentingNewLine]Table[{R, EnergyLevel[{R, 1}, {0, 1}, EMass \[Rule] {0.27, 0.27}]}, {R, 10, 60, 2}]\[IndentingNewLine], Table[{R, EnergyLevel[{R, 3.1}, {0, 1}, EMass \[Rule] {0.27, 0.27}]}, {R, 10, 60, 2}]\[IndentingNewLine], Table[{R, EnergyLevel[{R, 30}, {0, 1}, EMass \[Rule] {0.27, 0.27}]}, {R, 10, 60, 2}]\[IndentingNewLine]};\)\)], "Input", CellLabel->"In[50]:="], Cell[CellGroupData[{ Cell[BoxData[{ \(\(<< Graphics`MultipleListPlot`;\)\), "\n", \(\(MultipleListPlot[L\[IndentingNewLine], PlotJoined \[Rule] True\[IndentingNewLine], PlotRange \[Rule] All\[IndentingNewLine], Frame \[Rule] True\[IndentingNewLine], PlotLegend \[Rule] {"\", "\", "\"}\[IndentingNewLine], ImageSize \[Rule] 400\[IndentingNewLine], LegendPosition \[Rule] {0.2, 0}\[IndentingNewLine], LegendSize \[Rule] {0.6, 0.4}\[IndentingNewLine], FrameLabel \[Rule] {"\", "\"}\[IndentingNewLine], SymbolShape \[Rule] None\[IndentingNewLine], PlotLabel \[Rule] "\
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dependence of the same level is shown for \[Gamma] \[NonBreakingSpace]=\[NonBreakingSpace]1 and R\[NonBreakingSpace]= \[NonBreakingSpace]8\[NonBreakingSpace]\[CapitalARing], 1\[NonBreakingSpace] \[CapitalARing], 15\[NonBreakingSpace]\[CapitalARing], 30\[NonBreakingSpace] \[CapitalARing]. These figures verify that the energy increases when the radius decreases and when the barrier potential increases, which is a common feature of quantum confinement. Note that the dependence on this parameters is all the stronger that the radius and the barrier energy become smaller. To see the effect of the mass difference between the dot and the surrounding barrier we depict in Fig.3 the radius dependence of 1s-level for V0 \[NonBreakingSpace]=\[NonBreakingSpace]3.1\[NonBreakingSpace]eV \[Gamma] \[NonBreakingSpace]=\[NonBreakingSpace]0.5,1,2.5. 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We still \ see that at given R and V0 the energy is lower when \[Gamma] increases; \ moreover, the change of energy as a function of \[Gamma] is all the higher \ that V0 is smaller.\nTo estimate quantitatively the change of the electron \ energy depending on whether the difference of effective mass between \ materials is taken into account or not, we plot in Figs.5 and \ \[NonBreakingSpace]6 the quantity ", Cell[BoxData[ \(TraditionalForm\`E\_\(\[Gamma] = 1\)\)]], "-", Cell[BoxData[ \(TraditionalForm\`E\_\(\[Gamma] = 1.85\)\)]], " as a function of dot radius and barrier potential, respectively." }], "Text"], Cell[CellGroupData[{ Cell["graphics 5", "SubsubsectionIcon"], Cell[BoxData[ \(\(L = {\[IndentingNewLine]Table[{R, EnergyLevel[{R, 3.1}, {0, 1}, EMass \[Rule] {0.27, 0.27}] - EnergyLevel[{R, 3.1}, {0, 1}]}, {R, 10, 60, 2}]\[IndentingNewLine], Table[{R, EnergyLevel[{R, 3.1}, {1, 1}, EMass \[Rule] {0.27, 0.27}] - EnergyLevel[{R, 3.1}, {1, 1}]}, {R, 10, 60, 2}]\[IndentingNewLine], Table[{R, EnergyLevel[{R, 3.1}, {2, 1}, EMass \[Rule] {0.27, 0.27}] - EnergyLevel[{R, 3.1}, {2, 1}]}, {R, 10, 60, 2}]\[IndentingNewLine]};\)\)], "Input", CellLabel->"In[66]:="], Cell[CellGroupData[{ Cell[BoxData[{ \(\(<< Graphics`MultipleListPlot`;\)\), "\n", \(\(MultipleListPlot[L\[IndentingNewLine], PlotJoined \[Rule] True\[IndentingNewLine], PlotRange \[Rule] All\[IndentingNewLine], Frame \[Rule] True\[IndentingNewLine], PlotLegend \[Rule] {"\<1s\>", "\<2p\>", "\<3d\>"}\[IndentingNewLine], ImageSize \[Rule] 400\[IndentingNewLine], LegendPosition \[Rule] {0.2, 0}\[IndentingNewLine], LegendSize \[Rule] {0.6, 0.4}\[IndentingNewLine], FrameLabel \[Rule] {"\", "\"}\[IndentingNewLine], SymbolShape \[Rule] None\[IndentingNewLine], PlotLabel \[Rule] "\
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We can see clearly that the magnitude of the energy change is larger for small dots and small barrier energies. The typical energy change is about some tens meV for large dots and reaches more than 100\[NonBreakingSpace]meV for small dots, which cannot be neglected for most practical applications. 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If we use the conventional \ spectroscopic notation for the electron energy levels, we obtain an order of \ energy levels quite different from that of the hydrogen atom. The sequence is \ as follows\n\t", Cell[BoxData[ \(TraditionalForm\`E\_\(1 s\)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`E\_\(2 p\)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`E\_\(3 d\)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`E\_\(2 s\)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`E\_\(4 f\)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`E\_\(3 p\)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`E\_\(5 g\)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`E\_\(4 d\)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`E\_\(6 h\)\)]], " < ", Cell[BoxData[ \(TraditionalForm\`E\_\(3 s\)\)]], " ...\nThe difference in the order of levels between the hydrogen atom and \ the semiconductor QD is explained by the fact that in hydrogen atom the \ Coulomb potential is deeply singular at r\[NonBreakingSpace]=0, while there \ is no singularity in the dot. This sequence agrees with the results reported \ in Ref.11. We can also note that the separation of the successive energy \ levels increases when the numbernincreases, i.e.when we go up to the top of \ the dot well. This feature is quite contrary to the case of a normal atom, in \ which the separation between energy levels of highly excited states becomes \ smaller and smaller whennincreases. \nIn Fig.8 we can compare the energy \ structure in two cases: by including the material-dependence of mass in the \ calculation (dashed lines) or not (solid line). In accordance with results \ presented above, the energy levels resulting from a realistic model (\[Gamma] \ \[NonBreakingSpace]=1.85) are lower than using the simplified model (\[Gamma] \ \[NonBreakingSpace]=1). This situation may result in the emergence of new \ bound states which have been omitted in the simplified calculation. For \ example, the highly excited level 6d and 9k in Fig.8 are only observed for \ \[Gamma]\[NonBreakingSpace]=1.85. Additionally, the energy shift being \ significantly dependent on ", StyleBox["l", FontSlant->"Italic"], " value, one may observe a change in the sequence of levels (see e.g. 11n \ and 7g). By choosing the appropriate radius, barrier energy and mass ratio, \ one can expect this situation to happen for lower levels. \nFinally, we would \ like to compare our exact calculation with the results obtained from an \ approximate numerical method developed to treat many electron systems. Fig.9 \ presents the radius dependence of the first energy level calculated either by \ our exact treatment (solid line) or by self-consistently solving \ Poisson-Schr \[ODoubleDot]dinger equations using the Hartree approximation \ [10] (dashed line). In Fig.10 the corresponding wave functions resulting \ from both methods are shown. 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End of Graphics MathPictureEnd \ \>"], "Graphics", CellLabel->"From In[92]:=", ImageSize->{400, 247.188}, ImageCacheValid->False] }, Open ]] }, Closed]], Cell["Conclusion ", "Subsection"], Cell["\<\ We have presented the exact calculation of energy structure of a single electron in Si spherical nanocrystal surrounded by SiO2 forming a square well \ with finite barrier potential.Contrary to the case of simplified models used so far,the present model takes into account the material dependence of the electron effective mass.Fast and accurate calculations can be carried out to generate the energy structure for any combination of three inputs:radius of the dot,barrier potential of the dot and the ratio of barrier to dot electron \ effective mass.It is shown that the energy resulting from the simplified approach (with\[Gamma]\[NonBreakingSpace]=1) is overestimated for all states.The effect of material-dependence of electron effective mass becomes all the more significant that the quantum dot size and the barrier energy are \ smaller:the difference between realistic and simplified calculations of energy levels may reach more than 100 meV in small dots.Additionally,our calculation may result in new energy levels omitted by the simplified method and in a modified order of energy levels.As a future extension of this work,the basis of Bessel-like wavefunctions determined for a single electron could be generalised to treat the effect of Coulomb interaction in many-electron quantum dots.Such an approach should be faster to compute than the usual self-consistent Poisson-Schr\[ODoubleDot]dinger solver. \ \>", "Text"], Cell["Acknowledgement", "Subsection"], Cell["\<\ One of the authors (N.H.Quang) wishes to acknowledge financial support of the \ CNRS (PICS program).He is most appreciative for the hospitality shown to him while a visitor at the IEF,Orsay. \ \>", "Text"], Cell["References", "Subsection"], Cell["\<\ [1] T.Baron,P.Gentile,N.Magnea and P.Mur,Appl.Phys.Lett.79 (2001) 1175 [2] S.Tiwari,F.Rana,H.Hanafi,A.Hartstein and E.F.Crabb\[EAcute],Appl.Phys.Lett.68 (1996) 1377 [3] E.Kapetanakis,P.Normand,D.Tsoulakas,K.Beltsios,Appl.Phys.Lett.80 (2002) 2794 [4] S.Banerjee,Solid State Commun.125 (2003) 567 [5] R.Ohba,N.Sugiyama,K.Uchida,J.Koga and A.Toriumu,in Tech.Dig.IEDM'00 (2000) 313 [6] B.Choi,S.Hwang,I.Kim,H.Shin,Y.Kim and E.Kim,Appl.Phys.Lett.73 (1998) 3129 [7] Y.Niquet,C.Delerue,G.Allan and M.Lannoo,Phys.Rev.B,62 (2000) 5109 [8] A.Thean and J.Leburton,IEEE Trans.Electron Dev.,22 (2001) 148 [9] G.Iannacone and P.Coli,Appl.Phys.Lett.,78 (2001) 2046 [10] J.S\[EAcute]e,P.Dollfus and S.Galdin,J.Appl.Phys.,92 (2002) 3141 [11] Y.F.Zhao et.al.,J.Vac.Sci.Technol.B,15 (1997) 1452 [12] X.Leyronas,M.Combescot,Solid State Commun.119 (2001) 631; cond-mat/0201553 [13] Z.A.Weinberg,J.Appl.Phys.53 (1982) 5052 [14] E.Cassan,S.Galdin,P.Dollfus and P.Hesto,J.Appl.Phys.86 (1999) 3804 [15] J.S\[EAcute]e,P.Dollfus,and S.Galdin,Phys.Rev.B 66 (2002) 193307 [16] S.Wolfram,The 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