COULOMB BLOCKADE

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}]}};

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.+Sum[Sinh[alpha]/Sinh[n alpha],{n,2,100}];
N[4. Pi 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          = Pi 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]/e 1000/.caseC]//.NSub," [mV]"]
Print["dEc = ",N[dEc[C]/e 1000/.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/(C1+C2)*
(1/2 + pm(n - Q0/e) + pm C2 v/e)//.NSub;

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

Tunneling Rates

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

Clear[gamma1];
gamma2[n_,pm_,v_] :=
-dE2[n,pm,v]/(e^2 R2)*
If[dE2[n,pm,v]>0,0,1]//.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 = "(a)";
title2 = "(b)";
title3 = "(c)";
title4 = "(d)";
title5 = "(e)";

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]/e 1000]//.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 [mV]","i [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];