FOWLER-NORDHEIM 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

caseA
       = {phiAu -> 4.2 e, phiW -> 3.5 e, phiAl -> 3.17 e, 
          phiMg -> 2.3 e, phiSi -> 4.2 e, phiCr -> 3.51 e, 
          phiPt -> 3.01 e, Ecb -> 3.2 e} /. NSub;

case = Join[NSub, caseA];

Fowler-Nordheim Current Density

Clear[kappa]; kappa[phi_, meff_] := ((2 π (Sqrt[2 meff phi]))/h);

Clear[C0]; C0[phi_, meff_] := 
      ((2 (∫_0^phi) kappa[x, meff] \[DifferentialD]x)/e)

Print["C0 = ", C0[phi, meff]]

Clear[B0]; B0[phi_] := ((e^3)/(8 π h phi));

Clear[J]; J[v_, phi_, meff_, d_] = 
      B0[phi] ((((v/d)))^2) Exp[-((C0[phi, meff] d)/v)] /. case;

Numerical Example

Clear[Jx, phi]; Jx[phi_] := J[8, phi, 0.8 me, 3 nano] /. case

phix = 
      FindRoot[Jx[phi] == (10^5), {phi, 4 e /. NSub}]
          [[1, 2]]; Print["experimental phi = ", (phix/e) /. NSub, " [eV]"]

caseB = {phi -> phix, meff -> 0.8 me, d -> 3 nano};

case1 = Join[caseB, case];

B1 = B0[phi] /. case1; C1 = C0[phi, meff] /. case1;

Clear[J]; J[v_] := B1 ((((v/d)))^2) Exp[-((C1 d)/v)] /. case1;

Fig11x2 = 
      Plot[J[v]/(10^4), {v, 5, 8}, 
        FrameLabel -> {"v in V", "J in pA over 100 nm^2"}, 
        Evaluate[opt]];

Oxide Field and Applied Field

delphi = ((phi - phiSi)/e); Print["delphi = ", delphi /. caseB /. case, " [V]"]

Oscillation Factor

Clear[k1]; k1[v_] = (((((2 meff e v)/(d (hbar^2)))))^(1/3));

Print["k1[8] = ", N[k1[8]] nano //. case1, " [1/nm]"]

Clear[Lcb]; Lcb[v_, En_] := d - (((phi + En) d)/(e v));

Print["Lcb[8,0] = ", N[Lcb[8, 0]]/nano /. case1, " [nm]"]

Clear[k]; k[v_] = ((Sqrt[2 meff (phi + (e v - Ecb))])/hbar);

Print["k[8] = ", N[k[8] nano] //. case1, " [1/nm]"]

b0 = .08; Clear[b]; b[v_, En_] := 
      b0 N[1/(
                AiryAi[-k1[v] Lcb[v, En]]^2 + 
                  (((k1[v]/k[v]))^2) 
                    AiryAiPrime[-k1[v] Lcb[v, En]]^2)] //. case1;

Fig11x3 = 
      Plot[Evaluate[{b[v, 0], b[v, .2 e]}], {v, 5, 8}, 
        FrameLabel -> {"v in V", "b"}, Evaluate[opt]];

Clear[Josc]; Josc[v_, En_] := b[v, En] J[v];

Fig11x4 = 
      Plot[Evaluate[{J[v]/(10^4), Josc[v, 0]/(10^4)}], {v, 5, 8}, 
        FrameLabel -> {"v in V", "Josc in pA over 100 nm^2"}, 
        Evaluate[opt]];

Clear[G]; G[v_] := 
      -((v Log[Josc[v, 0]/(B1 ((((v/d)))^2))] nano)/d) /. case1

F[v_] := ((v nano)/d) /. case1

Fig11x5 = 
      ParametricPlot[{F[v], G[v]}, {v, 5, 8}, 
        FrameLabel -> {"F in V over nm", "G in V over nm"}, 
        Evaluate[opt]];

Print["C1 = ", N[nano C1 /. case1], " [nm/V]"]

Averaged Oscillations

Clear[C2]; C2[phi_, v_] := ((C1 d)/(phi v)) /. case1

Clear[bave]; bave[v_] := 
      C2[phi, v] 
        NIntegrate[
          b[v, En] Exp[-C2[phi, v] En], {En, 0, 
            e v - phiSi /. case1}];

tbl1 = Table[{v, bave[v]}, {v, 5, 10, .5}]; Fig11x6 = 
      ListPlot[tbl1, PlotJoined -> True, 
        FrameLabel -> {"v in V", "bave"}, Evaluate[opt1]];

Effective Tunneling Area