DENSITY OF STATES

Remove["Global`*"]

Highlights

Abstract

Introduction

Off[General::spell,General::spell1];

Needs["Graphics`Arrow`"];

tera  = 10^12  ; micro = 10^-6;
giga  = 10^9   ; nano  = 10^-9; 
mega  = 10^6   ; pico  = 10^-12;
kilo  = 10^3   ; femto = 10^-15; 
milli = 10.^-3 ; atto  = 10^-18;

NSub = {
c   -> 2.99792458 10^8  , e     -> 1.6021773 10^-19,
h   -> 6.6260755 10.^-34, hbar  -> 1.05457 10.^-34 ,
K   -> 1.380658 10^-23  , me    -> 9.109389 10^-31 ,
mu0 -> 4. Pi 10^-7      , eps0  -> 8.85419 10^-12   };

ESi   =0.179 tera; rhoSi   =2330 ; nuSi   =0.28;
ESiO2 =0.06  tera; rhoSiO2 =2200 ; nuSiO2 =0.3;
ESi3N4=0.15  tera; rhoSi3N4=3100 ; nuSi3N4=0.24;
EAu   =0.08  tera; rhoAu   =19300; nuAu   =0.42;
ENi   =0.207 tera; rhoNi   =8900 ; nuNi   =0.41;
EW    =0.345 tera; rhoW    =19300; nuW    =0.28;
EPt   =0.17  tera; rhoPt   =21400; nuPt   =0.39;
EIr   =0.52  tera; rhoIr   =22500; nuIr   =0.3 ;
EHOPG =0.2   tera; rhoHOPG =2250 ; nuHOPG =0.3 ;
EPol  =0.001 tera; rhoPol  =1000 ; nuPol  =0.3 ;

xx     =0.8;
EPtIr  =xx 0.8 EPt+(1-xx) EIr;
rhoPtIr=xx rhoPt  +(1-xx) rhoIr;
nuPtIr =0.3;

opt= {
PlotRange  ->All,
AxesOrigin ->{0,0},
Frame      ->True,
GridLines  ->Automatic,           
DefaultFont->{"Times-Roman",8},
PlotStyle  ->{
Dashing[{.0001}],
Dashing[{.0200}],
Dashing[{.0200}],
Dashing[{.0200}],
Dashing[{.0200}],
Dashing[{.0200}]}};

case1 = {L->1 nano};
case  = Join[NSub,case1];

Sphere in Arbitrary Dimensions

Volume

Clear[Vs];
Vs[d_,r_] := r^d Pi^(d/2) 1/Gamma[1+d/2];

tbl1 = Table[Vs[d,r],{d,0,3,1}]

Fig14x1 = Plot[{
Vs[d,1.1],
Vs[d,1.2]},
{d,0,20},
FrameLabel ->{"d","V(d,r)"},
Evaluate[opt]];

Clear[dVs];
dVs[d_,r_] = D[Vs[d,r],{d,1}];
Clear[damx];
dmax[r_] := FindRoot[dVs[d,r],{d,5}][[1,2]]
Print["dmax = ",dmax[1.1]]

Area

Clear[As];
As[d_,r_] := D[Vs[d,r],r];

tbl2 = Table[As[d,r],{d,0,3,1}]

Solid Angle

Clear[thetas,r];
thetas[d_] = D[Vs[d,r],r]/r^(d-1);

tbl3 = Table[thetas[d],{d,0,3,1}]

Density of States in Arbitrary Dimensions

Volume in k-Space

Clear[Vk,k];
Vk[d_,k_] := Vs[d,r]/.r->k

tbl4 = Table[Vk[d,k],{d,1,3,1}]

Energy in the Parabolic Approximation

Clear[En,k];
En[k_] := hbar^2 k^2/(2 me);

kEn = Solve[En[k]==En,k][[2,1,2]];
Print["kEn = ",kEn]

Density of States

Clear[nk];
nk[d_,k_] := 2 1/(2 Pi)^d Vk[d,k]

Clear[nkEn];
nkEn[d_,En_] := nk[d,k]/.k->kEn;

Clear[rho];
rho[d_,En_] = D[nkEn[d,En],En];

Print["dos in 1-d = ",rho[1,En]]
Print["dos in 2-d = ",rho[2,En]]
Print["dos in 3-d = ",rho[3,En]]

Fig14x2 = Plot[{
atto L rho[1,e En]/.case,
atto L^2 rho[2,e En]/.case,
atto L^3 rho[3,e En]/.case},
{En,0.01,2},
AxesOrigin ->{0,0},
FrameLabel ->{"E [eV]","DOS"},
Evaluate[opt]];

Fig14x3 = Plot3D[
atto Vs[d,L] rho[d,e En]/.case,
{d,0,3},{En,0.1,2},
DefaultFont->{"Times-Roman",8},
AxesLabel  ->{"d","E [eV]","DOS"}];

Density of States in Confined Structures

Quantum Well

Clear[rho10,rho11];
rho10[En_,s_] := L/s rho[2,En]
rho11[En_,s_] := rho10[En,s]*
Floor[L rho[3,En]/rho10[En,s]]

Fig14x4 = Plot[Evaluate[{
atto L^3 rho[3,e En]//.case,
atto L^2 rho11[e En,2 nano]//.case}],
{En,0.01,2},
FrameLabel ->{"E [eV]","DOS"},
Evaluate[opt]];

Quantum Wire

Clear[rho20,rho22];
rho20[En_,s_] := (L/s)^2 rho[1,En]
rho22[En_,s_] := rho20[En,s]*
Floor[L^2 rho[3,En]/rho20[En,s]]

Fig14x5 = Plot[Evaluate[{
atto L^3 rho[3,e En]//.case,
atto L rho22[e En,1 nano]//.case}],
{En,0.01,2},
FrameLabel ->{"E [eV]","DOS"},
Evaluate[opt]];

Cubical Quantum Dot

Clear[Ec];
Ec[Lc_,n_] := 3 En[k]/.k->n Pi/Lc;

tbl5 = Table[Ec[Lc nano,n]/e,{n,1,6,1}];

Fig14x6 = Plot[Evaluate[
tbl5/.case],
{Lc,1,5},
FrameLabel ->{"L [nm]","Ec [eV]"},
Evaluate[opt]];

Spherical Quantum Dot

tbl6 = {Pi,4.4934,5.7635,2 Pi,7.7253,9.0950};

Clear[Es];
Es[R_] := hbar^2/(2 me) tbl6^2/R^2;

Fig14x7 = Plot[
Evaluate[Es[R nano]/e/.case],
{R,1,5},
FrameLabel ->{"R [nm]","Es [eV]"},
Evaluate[opt]];

Fig14x8 = Plot[Evaluate[{
Es[Lc nano][[1]]/e/.case,
Ec[Lc nano (4/3 Pi)^(1/3),1]/e/.case}],
{Lc,1,3},
FrameLabel ->{"L [nm]","En [eV]"},
Evaluate[opt]];