DENSITY OF STATES

Remove["Global`*"]

Highlights

Abstract

Introduction

Off[General::"spell", General::"spell1"]; Needs["Graphics`Arrow`"]; tera = (10^12); micro = (1/(10^6)); giga = (10^9); nano = (1/(10^9)); mega = (10^6); pico = (1/(10^12)); kilo = (10^3); femto = (1/(10^15)); milli = (1/(10.^3)); atto = (1/(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. π)/(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[{.0001}]}};

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

Sphere in Arbitrary Dimensions

Volume

Clear[Vs]; Vs[d_, r_] := (((r^d) (π^(d/2)))/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"}, Evaluate[opt]];

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

Area

Clear[As]; As[d_, r_] := (∂_r) Vs[d, r];

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

Solid Angle

Clear[thetas, r]; thetas[d_] = (∂_r) Vs[d, 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 Vk[d, k])/(((2 π))^d))

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

Clear[rho]; rho[d_, En_] = (∂_En) nkEn[d, 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 in 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 rho[2, En])/s)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 in 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 in eV", "DOS"}, Evaluate[opt]];

Cubical Quantum Dot

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

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

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

Spherical Quantum Dot

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

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

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

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