METAL-INSULATOR-METAL TUNNELING

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}]}}; 

    \(opt1 = {PlotRange -> All, AxesOrigin -> {0, 0}, 
        Frame -> True, GridLines -> Automatic, 
        DefaultFont -> {"Times-Roman", 8}}; \)\)

Operating Conditions

case = {phi -> 4 e, epsr -> 5.} /. NSub;

case1 = Join[case, NSub];

Nphi = (phi/e) /. case1;

Tunneling Current Density

General Solution

Clear[A, J0, Jg, NJg]; A[ds_] := ((4 π (Sqrt[2 me]) ds)/h); J0[ds_] := (e/(2 π h (ds^2))); Jg[v_, ds_, phibar_] := 
      J0[ds] (
          phibar Exp[-A[ds] (Sqrt[phibar])] - 
            (phibar + e v) Exp[-A[ds] (Sqrt[phibar + e v])]);

NJg[v_, ds_, phibar_] := N[Jg[v, ds, phibar] /. case1]

Small Voltage Approximation

Clear[Jgsmall, NJgsmall]; Jgsmall[v_, s_] := Jg[v, s, phi - ((e v)/2)]; NJgsmall[v_, s_] := N[Jgsmall[v, s] /. case1];

Clear[Jsmall, NJsmall]; Jsmall[v_, s_] := 
      (((Sqrt[2 me phi]) ((((e/h)))^2) v 
          Exp[-((4 π s (Sqrt[2 me phi]))/h)])/s); NJsmall[v_, s_] := N[Jsmall[v, s] /. case1]

stest = 1 nano;

Fig10x2 = 
      Plot[Evaluate[{micro NJgsmall[v, stest], micro NJsmall[v, stest]}], {
          v, .01, 1}, FrameLabel -> {"v in V", "J in MA/m^2"}, 
        Evaluate[opt]];

vtest = 1.;

Fig10x3 = 
      Plot[Evaluate[{micro NJgsmall[vtest, s nano], 
            micro NJsmall[vtest, s nano]}], {s, .9, 1.1}, 
        FrameLabel -> {"s in nm", "J in MA/m^2"}, 
        Evaluate[opt]];

Large Voltage Approximation

Clear[Jg1arge, NJglarge]; Jglarge[v_, s_] := Jg[v, ((s phi)/(e v)), (phi/2)]; NJglarge[v_, s_] := N[Jglarge[v, s] /. case1];

Clear[Jlarge, NJlarge]; Jlarge[v_, s_] := 
      (((e^3) (F^2) 
            Exp[-((8 π phi (Sqrt[2 me phi]))/(3 h e F))]
              )/(8 π h phi)) /. F -> (v/s); NJlarge[v_, s_] := N[Jlarge[v, s] /. case1]

stest = 1 nano;

Fig10x4 = 
      Plot[Evaluate[{micro NJglarge[v, stest], micro NJlarge[v, stest]}], {
          v, 4, 6}, FrameLabel -> {"v in V", "J in MA/m^2"}, 
        Evaluate[opt]];

vtest = 5.;

Fig10x5 = 
      Plot[Evaluate[{micro NJglarge[vtest, s nano], 
            micro NJlarge[vtest, s nano]}], {s, .9, 1.1}, 
        FrameLabel -> {"s in nm", "J in MA/m^2"}, 
        Evaluate[opt]];

The Image Potential

g = ((1.15 e Log[2])/(8 π eps0 epsr)) /. case1; Print["g = ", N[g]/nano, " [V nm]"]

Clear[vi]; vi[x_] := -((g s)/(x (s - x)));

Print["vi[s/2] = ", N[vi[(s/2)]] /. s -> 1 nano, " [V]"]

Barrier with an Image Potential

Clear[phii]; phii[v_, s_, x_] := 
      phi - ((e v x)/s) - ((e g1 s)/(x (s - x)));

Clear[Nphii]; Nphii[v_, s_, x_] := 
      N[phii[v, s, x]/e /. case1 /. g1 -> g]

Clear[Nphii1]; Nphii1[v_, s_, x_] := If[x < s, Nphii[v, s, x], 0]

stest = 1 nano;

Fig10x6 = 
      Plot[Evaluate[{Nphii1[0, stest, x nano], Nphii1[2, stest, x nano], 
            Nphii1[4, stest, x nano], Nphii1[6, stest, x nano]}], {
          x, .01, .99}, PlotRange -> {0, Nphi}, 
        FrameLabel -> {"v in V", "phi in eV"}, Evaluate[opt]];

vtest = 5.;

Fig10x7 = 
      Plot[Evaluate[{Nphii1[vtest, .8 nano, x nano], 
            Nphii1[vtest, .9 nano, x nano], 
            Nphii1[vtest, 1. nano, x nano], 
            Nphii1[vtest, 1.1 nano, x nano]}], {x, .01, .99}, 
        PlotRange -> {0, Nphi}, 
        FrameLabel -> {"x in nm", "phi in eV"}, Evaluate[opt]];

The Barrier Width

\(Clear[xi, xi1, xi2, dxi]; 

    xi1[v_, s_] := \(Solve[Nphii[v, s, x] == 0, x]\)[\([1, 1, 2]\)]; 

    xi2[v_, s_] := \(Solve[Nphii[v, s, x] == 0, x]\)[\([2, 1, 2]\)]; 

    xi3[v_, s_] := \(Solve[Nphii[v, s, x] == 0, x]\)[\([3, 1, 2]\)]; 

    dxi[v_, s_] := xi2[v, s] - xi1[v, s]; \)

stest = 1 nano;

Fig10x8 = 
      Plot[dxi[v, stest]/nano, {v, .01, 5}, 
        AxesOrigin -> {0, (stest/nano)}, 
        FrameLabel -> {"v in V", "dxi in nm"}, Evaluate[opt]];

vtest = 5.;

Fig10x9 = 
      Plot[dxi[vtest, x nano]/nano, {x, .9, 1.1}, 
        FrameLabel -> {"s in nm", "dxi in nm"}, Evaluate[opt]];

Average Barrier Height

Clear[phibari]; phibari[v_, s_] := 
      (((∫_(xi1[v, s])^(xi2[v, s]))Nphii[v, s, x] 
            \[DifferentialD]x)/dxi)[v, s];

stest = 1 nano; tbl1 = Table[{v, Re[phibari[v, stest]]}, {v, .01, 5.01, 1}];

Fig10x10 = 
      ListPlot[tbl1, AxesOrigin -> {0, 0}, PlotRange -> {0, Nphi}, 
        PlotJoined -> True, 
        FrameLabel -> {"v in V", "phibari in eV"}, 
        Evaluate[opt1]];

vtest = 5.; tbl2 = Table[{s, Re[phibari[vtest, s nano]]}, {s, .9, 1.1, .05}];

Fig10x11 = 
      ListPlot[tbl2, AxesOrigin -> {0, 0}, PlotJoined -> True, 
        FrameLabel -> {"s in nm", "phibari in eV"}, 
        Evaluate[opt1]];

Comparison of the Barriers

stest = 1 nano; vtest = 5.;

s0 = (stest/nano); \(xa = xi1[vtest, stest]\/nano; 

xb = xi2[vtest, stest]\/\(nano
	\); 
    
v1 = vtest; 
\(p1 = Re[phibari[vtest, stest]]; \)\)p = Nphi;

Barrier without an Image Potential

barrierGraphicsA = 
      Show[Graphics[{
            Line[{{-1, 0}, {0, 0}, {0, p}, {s0, p - v1}, {s0, -v1}, {
                  s0 + 1, -v1}}], 
            Text[FontForm["metal", {"Times-Roman", 8}], {-.5, 
                -3}], 
            Text[FontForm["phi", {"Times-Roman", 8}], {-.2, 
                p + .2}], 
            Text[FontForm["-v", {"Times-Roman", 8}], {xb - .1, 
                -v1}], 
            Text[FontForm["phi-v", {"Times-Roman", 8}], {s0 + .3, 
                p - v1}], 
            Text[FontForm["insulator", {"Times-Roman", 8}], {(s0/2), 
                -3}], 
            Text[FontForm["metal", {"Times-Roman", 8}], {s0 + .5, 
                -3}]}], Frame -> True, 
        PlotRange -> {{-1, 2}, {-7, 5}}, 
        DisplayFunction -> Identity];

Barrier with an Image Potential

barrierGraphicsB = 
      Plot[Evaluate[Nphii[v1, stest, x nano]], {x, .001, s0 - .001}, 
        Frame -> True, PlotRange -> {-6, 4}, 
        AxesOrigin -> {-1, -6}, 
        DefaultFont -> {"Times-Roman", 8}, 
        PlotStyle -> Dashing[{.01}], DisplayFunction -> Identity];

Average Barrier Height

barrierGraphicsC = 
      Show[Graphics[{
            Line[{{0, 0}, {xa, 0}, {xa, p1}, {xb, p1}, {xb, -v1}, {
                  xb + 1, -v1}}], 
            Text[FontForm["phibar", {"Times-Roman", 8}], {.35, 
                p1 - .6}]}], DisplayFunction -> Identity];

The Plot of the Three Barriers

Fig10x12 = 
      Show[{barrierGraphicsA, barrierGraphicsB, barrierGraphicsC}, 
        Frame -> True, DefaultFont -> {"Times-Roman", 8}, 
        FrameLabel -> {"s in nm", "phi in V"}, 
        DisplayFunction -> $DisplayFunction];

Comparison of Tunneling Currents

The General Solution with Image Potential

Clear[Jgi]; Jgi[v_, s_] := NJg[v, dxi[v, s], Re[e phibari[v, s]]];

stest = 3 nano; tbl3 = Table[{v, N[Log[10, (v/Jgi)[v, stest]]]}, {v, .01, 6.01, 1}];

Small Voltage Approximation without Image Potential

tbl4 = 
      Table[{v, Log[10, N[(v/NJgsmall)[v, stest]]]}, {v, .01, 1.01, .05}];

Large Voltage Approximation without Image Potential

tbl5 = Table[{v, Log[10, (v/NJlarge)[v, stest]]}, {v, 4, 6, .05}];

Plot of the Three Tunneling Currents

resGraphicsA = 
      ListPlot[tbl3, PlotJoined -> True, 
        FrameLabel -> {"v in V", "log v over i in Ohm m^2"}, 
        DisplayFunction -> Identity, Evaluate[opt1]];

resGraphicsB = 
      ListPlot[tbl4, PlotJoined -> True, 
        PlotStyle -> Dashing[{.02}], DisplayFunction -> Identity, 
        Evaluate[opt1]];

resGraphicsC = 
      ListPlot[tbl5, PlotJoined -> True, 
        PlotStyle -> Dashing[{.02}], DisplayFunction -> Identity, 
        Evaluate[opt]];

Fig10x13 = 
      Show[{resGraphicsA, resGraphicsB, resGraphicsC}, 
        DisplayFunction -> $DisplayFunction];

Apparent Barrier Height

kappa = N[((2 π (Sqrt[2 me phi]))/h) /. case1]; Print["kappa = ", (kappa/(10^10)), " [1/Angstrom]"]

a = ((2 ((((h/(4 π))))^2) (10^20))/(me e)) /. case1; Print["a = ", N[a], " [eV Angstrom^2]"]

Clear[dlnJgi]; dlnJgi[v_, s_] := ((Jgi[v, s + ds]/Jgi[v, s] - 1.)/ds);

ds = .01 nano;

A0 = N[((4 π (Sqrt[2 me]))/h) /. case1]; stest = 1 nano;

tbl6 = 
      Table[{v, ((((dlnJgi[v, stest]/A0))^2)/e) /. case1}, {v, .01, 5.01, 
          1}];

\(Fig10x14 = 
      ListPlot[tbl6, PlotRange -> {0, Nphi}, PlotJoined -> True, 
        FrameLabel -> {"v in V", "phiapp in V"}, 
        DisplayFunction -> Identity, Evaluate[opt]]; 

    \(Show[{Fig10x14, Fig10x10}, DisplayFunction -> $DisplayFunction]; 
    \)\)

vtest = 1.;

tbl7 = 
      Table[{s, ((((dlnJgi[vtest, s nano]/A0))^2)/e) /. case1}, {s, .9, 
          1.1, .1}];

Fig10x15 = 
      ListPlot[tbl7, PlotJoined -> True, 
        FrameLabel -> {"s in nm", "phiapp in V"}, 
        Evaluate[opt1]];