COULOMB BLOCKADE

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[{.0200}]}};

Capacitance

Sphere-Plane Capacitance

Clear[Csp, R, s]; Csp := 
      Function[{R, 
          s}, {alpha = Log[1. + (s/R) + (Sqrt[((s^2)/(R^2)) + ((2. s)/R)])]; 
            cap2 = 1. + (∑_(n = 2)^100) Sinh[alpha]/Sinh[n alpha]; 
            N[4. π eps0 R cap2]}[[1
          ]]]

caseA = {R -> 100 nano, s -> 50 nano}; Print["Csp = ", Csp[R, s]/atto /. caseA /. NSub, " [aF]"]

Sphere-Sphere Capacitance

Clear[Cssa, R1, R2, s]; Cssa := 
      Function[{R1, R2, 
          s}, {R11 = Max[{R1, R2}]; R22 = Min[{R1, R2}]; 
            rb = (1/(1 + (R22/R11))); N[rb Csp[R22, rb s]] //. NSub}
            [[1]]]

caseB
       = {R1 -> 100 nano, R2 -> 100 nano, s -> 100 nano}; 
    Print["Cssa = ", Cssa[R1, R2, s]/atto /. caseB, " [aF]"]

Quantum Considerations

Clear[Eth, dEc, tau, VQ, nu]; RQ = ((π hbar)/(2 (e^2))) /. NSub; Eth[T_] := K T /. NSub; dEc[C_] := ((e^2)/(2 C)) /. NSub; tau[R_, C_] := R C /. NSub; VQ[R_, C_] := (hbar/(e R C)) /. NSub; nu[i_] := (i/e) /. NSub;

Requirements and Approximations

caseC
       = {T -> 300, C -> 1 atto, R -> (10^7), 
        i -> 1 nano};

Print["RQ  = ", N[(RQ/1000)], " [kOhm]"]; Print["nu  = ", N[nu[i] nano /. caseC], " [GHz]"]; Print["tau = ", N[tau[R, C]/pico /. caseC], " [psec]"]Print["Eth = ", N[((Eth[T] 1000)/e) /. caseC] //. NSub, 
      " [mV]"]Print["dEc = ", N[((dEc[C] 1000)/e) /. caseC] //. NSub, 
      " [mV]"]Print["VQ  = ", N[VQ[R, C] 1000 /. caseC], " [mV]"]

Coulomb Blockade and Coulomb Staircase

Electrostatic Energy due to the Charging of the Center Electrode

Electrostatic Energy due to the Applied Bias

Total Electrostatic Energy

Clear[dE1]; dE1[n_, pm_, v_] := 
      (((e^2) ((1/2) + pm (n - (Q0/e)) + ((pm C2 v)/e)))/(C1 + C2
            )) //. NSub;

Clear[dE2]; dE2[n_, pm_, v_] := 
      (((e^2) ((1/2) + pm (n - (Q0/e)) - ((pm C1 v)/e)))/(C1 + C2
            )) //. NSub;

Tunneling Rates

Clear[gamma1]; gamma1[n_, pm_, v_] := 
      -((dE1[n, pm, v] If[dE1[n, pm, v] > 0, 0, 1])/((e^2) R1
                )) //. NSub;

Clear[gamma1]; gamma2[n_, pm_, v_] := 
      -((dE2[n, pm, v] If[dE2[n, pm, v] > 0, 0, 1])/((e^2) R2
                )) //. NSub;

Tunneling Current

Clear[i0, i]; i0[n_, v_] := 
      e (gamma2[n, 1, v] - gamma2[n, -1, v]) //. NSub;

Clear[i]; i[v_] := i0[Floor[((-C2 v + Q0 + (e/2))/e)], v] //. NSub

Examples

Clear[T, title, imax, Q0, C1, C2, R1, R2];

\(title1 = "case1"; 
title2 = "case2"; 
title3 = "case3"; 

    title4 = "case4"; 
title5 = "case5"; \)

case1
       = {title -> title1, T -> 4.2, imax -> 1.2 nano, 
        Q0 -> -.096 e, C1 -> 13.6 atto, R1 -> .3 (10^6), 
        C2 -> 4.05 atto, R2 -> 29.3 (10^6)};

case2
       = {title -> title2, T -> 4.2, imax -> 750 pico, 
        Q0 -> -.005 e, C1 -> 1.64 atto, R1 -> 2 (10^6), 
        C2 -> 3.28 atto, R2 -> 39.2 (10^6)};

case3
       = {title -> title3, T -> 4.2, imax -> 10.9 nano, 
        Q0 -> -.119 e, C1 -> 0.938 atto, 
        R1 -> 1.7 (10^6), C2 -> 0.734 atto, 
        R2 -> 16.6 (10^6)};

case4
       = {title -> title4, T -> 4.2, imax -> 550 pico, 
        Q0 -> 0 e, C1 -> 2 atto, R1 -> 34.9 (10^6), 
        C2 -> 4.14 atto, R2 -> 132 (10^6)};

case5
       = {title -> title5, T -> 4.2, imax -> 65 pico, 
        Q0 -> 0 e, C1 -> 1 atto, R1 -> 1 (10^6), 
        C2 -> 100 atto, R2 -> 100 (10^6)};

Model Calculations

case = case5;

{T, title, imax, Q0, R1, R2, C1, C2}
         = {T, title, imax, Q0, R1, R2, C1, C2} //. case; vmax = imax (R1 + R2); n1 = Floor[((-C2 vmax + Q0 + (e/2))/e)] //. NSub; n2 = Ceiling[((-C2 vmax + Q0 - (e/2))/e)] //. NSub; nmax = Max[{n1, n2}];

Print["nmax = ", Abs[nmax], " [e]"]; Print["tau  = ", N[Max[{R1 C1, R2 C2}]/nano], " [nsec]"]; 
    Print["Eth  = ", N[((1000 K T)/e)] //. NSub, " [mV]"]; 
    Print["dEc  = ", N[((dEc[C1 + C2] 1000)/e)] //. NSub, 
      " [mV]"]; Print["VQ   = ", N[VQ[R1, C1] 1000], " [mV]"];

Fig13x2 = 
      Plot[i[(v/1000)]/nano /. case, {v, -vmax 1000, vmax 1000}, 
        PlotRange 
          -> {{-vmax 1000, vmax 1000}, {-(imax/nano), 
              (imax/nano)}}, Frame -> True, 
        DefaultFont -> {"Times-Roman", 8}, 
        FrameLabel -> {"v in mV", "i in nA"}, 
        Epilog -> 
          Text[FontForm[title, {"Times-Roman", 8}], {-vmax 900, 
              ((.9 imax)/nano)}]];

Data Analysis

e = e /. NSub; sol = 
      NSolve[{((-(e/2) + QQ)/CC2) == -.135, 
          ((-(e/2) - QQ)/CC1) == -.065, 
          (((e/2) + QQ)/CC2) == .083}, {QQ, CC1, CC2}];

Print["C1 = ", 
      sol[[1, 1, 2]]/atto, 
      " [aFarad]"]Print["C2 = ", 
      sol[[1, 2, 2]]/atto, 
      " [aFarad]"]Print["Q0 = ", 
      sol[[1, 3, 2]]/e, " [e]"]Clear[e];