GeFiCa Germanium detector Field Calculator
analytic/getVd.cc
// definition of necessary units
static const double cm=1;
static const double cm3=cm*cm*cm;
static const double volt=1;
static const double C=1; // Coulomb
static const double Qe=-1.6e-19*C; // eletron charge
static const double epsilon0=8.854187817e-14*C/volt/cm; // vacuum permittivity
static const double epsilonGe=15.8; // Ge dielectric constant
//______________________________________________________________________________
// V"(x)=a, https://www.wolframalpha.com/input/?i=V%27%27(x)%3Da
double V(double *coordinates, double *parameters)
{
double x = coordinates[0]; // there is no y and z dependence
double x0= 0*cm; // lower electrode
double x1= parameters[0]; // upper electrode
double v0= 0*volt; // lower voltage
double v1= parameters[1]; // upper voltage
double rho=parameters[2]*Qe;// space charge density [C/cm3]
double a =-rho/epsilon0/epsilonGe;
double c2= (v1-v0)/(x1-x0) - a/2*(x1+x0);
double c1= (v0*x1-v1*x0)/(x1-x0) + a/2*x0*x1;
return a*x*x/2 + c2*x + c1;
}
//______________________________________________________________________________
// search for depletion voltage of a planar detector given impurity & thickness
double GetVdep(double impurity, double thickness, double vupper)
{
if (impurity==0) return 0; // nothing to deplete
TF1 *det=new TF1("det", V, 0, thickness, 3); // potential distr.
double bias, vlower=0*volt; // range of search
while (abs(vupper-vlower)>1e-3*volt) { // binary search
bias=(vupper+vlower)/2; // bias voltage
det->SetParameters(thickness, bias, impurity);
if (det->Derivative(0)*det->Derivative(thickness)<0) vlower=bias;
else vupper=bias;
}
return bias;
}
//______________________________________________________________________________
// draw results
void getVd(int type=1, double thickness=1*cm)
{
const int n=10; // number of points
double impurity[n]={1e9/cm3, 2e9/cm3, 4e9/cm3, 8e9/cm3, 1e10/cm3,
2e10/cm3, 4e10/cm3, 8e10/cm3, 1e11/cm3, 1.2e11/cm3}; // p-type
if (type==-1) for (int i=0; i<n; i++) impurity[i]*=type; // n-type
double absi[n], vdep[n];
for (int i=0; i<n; i++) {
vdep[i] = GetVdep(impurity[i], thickness, type*-2e4*volt);
absi[i] = abs(impurity[i]);
}
gROOT->SetStyle("Plain"); // pick up a good default drawing style to modify
// modify the default style
gStyle->SetLegendBorderSize(0);
gStyle->SetLegendFont(132);
gStyle->SetLabelFont(132,"XY");
gStyle->SetTitleFont(132,"XY");
gStyle->SetLabelSize(0.05,"XY");
gStyle->SetTitleSize(0.05,"XY");
gStyle->SetTitleOffset(1.1,"XY");
// Vdep VS impurity
TGraph *g = new TGraph(n,absi,vdep);
g->SetTitle(";Net Impurity [cm^{-3}];Depletion Voltage [V]");
g->Draw("apc");
TText *t1 = new TText(2e9, 6000,
Form("%.0f cm thick planar detector", thickness/cm));
t1->Draw();
// voltage VS thickness
double bias = -1000*volt*type;
TCanvas *c = new TCanvas;
// over depleted
TF1 *fvo=new TF1("fvo", V, 0, thickness, 3);
fvo->SetParameters(thickness, vdep[6]+bias, impurity[6]);
fvo->SetTitle(";Vertical position in planar detector [cm]; Voltage [V]");
fvo->SetLineColor(kMagenta);
fvo->SetLineStyle(2);
fvo->GetXaxis()->SetTitleOffset(1.1);
fvo->Draw();
// depleted
TF1 *fvd=new TF1("fvd", V, 0, thickness, 3);
fvd->SetParameters(thickness, vdep[6], impurity[6]);
fvd->Draw("same");
// undepleted
TF1 *fvu=new TF1("fvu", V, 0, thickness, 3);
fvu->SetParameters(thickness, bias, impurity[6]);
fvu->SetLineColor(kRed);
fvu->SetLineStyle(3);
fvu->Draw("same");
// potential due to bias alone
TF1 *fvb=new TF1("fvb", V, 0, thickness, 3);
fvb->SetParameters(thickness, bias, 0/cm3);
fvb->SetLineColor(kBlue);
fvb->SetLineStyle(4);
fvb->Draw("same");
// potential due to space charges alone
TF1 *fvc=new TF1("fvc", V, 0, thickness, 3);
fvc->SetParameters(thickness, 0*volt, impurity[6]);
fvc->SetLineColor(kGreen);
fvc->SetLineStyle(5);
fvc->Draw("same");
if (type==-1) { // draw lines and texts for n-type
TLine *l1 = new TLine(0,bias/volt,thickness/cm,bias/volt);
l1->SetLineStyle(kDashed); l1->Draw();
TLine *l2 = new TLine(0,vdep[6]/volt,thickness/cm,vdep[6]/volt);
l2->SetLineStyle(kDashed); l2->Draw();
TLine *l3=new TLine(0,(vdep[6]+bias)/volt,thickness/cm,(vdep[6]+bias)/volt);
l3->SetLineStyle(kDashed); l3->Draw();
TText *t2 = new TText(0.04,bias/volt+20,Form("%.0f V", bias/volt));
t2->SetTextFont(132); t2->Draw();
TText *t3 = new TText(0.04,vdep[6]/volt+20, Form("%.0f V", vdep[6]/volt));
t3->SetTextFont(132); t3->Draw();
TText *t4 = new TText(0.04,(vdep[6]+bias)/volt+20,
Form("%.0f V", (vdep[6]+bias)/volt));
t4->Draw();
TText *t5 = new TText(0.5, 2800, "over depleted");
t5->SetTextFont(132); t5->SetTextColor(kMagenta); t5->Draw();
TText *t6 = new TText(0.8, 2310, "just depleted");
t6->SetTextFont(132); t6->Draw();
TText *t7 = new TText(0.65, 1250, "undepleted");
t7->SetTextFont(132); t7->SetTextColor(kRed); t7->Draw();
TText *t8 = new TText(0.65, 800, "bias alone");
t8->SetTextFont(132); t8->SetTextColor(kBlue); t8->Draw();
TText *t9 = new TText(0.55, 200, "space charges alone");
t9->SetTextFont(132); t9->SetTextColor(kGreen); t9->Draw();
TLatex *t10 = new TLatex(0.1, 2700,
Form("Impurity: %.0e/cm^{3}",impurity[6]/cm3));
t10->SetTextFont(132); t10->Draw();
} else { // draw lines and texts for p-type
TLine *l0 = new TLine(0,0,thickness/cm,0);
l0->SetLineStyle(kDashed); l0->Draw();
TLine *l1 = new TLine(0,bias/volt,thickness/cm,bias/volt);
l1->SetLineStyle(kDashed); l1->Draw();
TLine *l2 = new TLine(0,vdep[6]/volt,thickness/cm,vdep[6]/volt);
l2->SetLineStyle(kDashed); l2->Draw();
TLine *l3=new TLine(0,(vdep[6]+bias)/volt,thickness/cm,(vdep[6]+bias)/volt);
l3->SetLineStyle(kDashed); l3->Draw();
TText *t2 = new TText(0.04,bias/volt+20,Form("%.0f V", bias/volt));
t2->SetTextFont(132); t2->Draw();
TText *t3 = new TText(0.04,vdep[6]/volt+20, Form("%.0f V", vdep[6]/volt));
t3->SetTextFont(132); t3->Draw();
TText *t4 = new TText(0.04,(vdep[6]+bias)/volt+20,
Form("%.0f V", (vdep[6]+bias)/volt));
t4->SetTextFont(132); t4->Draw();
TText *t5 = new TText(0.76, -2850, "over depleted");
t5->SetTextFont(132); t5->SetTextAngle(-19); t5->SetTextColor(kMagenta);
t5->Draw();
TText *t6 = new TText(0.8, -2120, "just depleted");
t6->SetTextFont(132); t6->SetTextAngle(-9); t6->Draw();
TText *t7 = new TText(0.65, -1350, "undepleted");
t7->SetTextFont(132); t7->SetTextColor(kRed); t7->Draw();
TText *t8 = new TText(0.75, -700, "bias alone");
t8->SetTextFont(132); t8->SetTextAngle(-11.5); t8->SetTextColor(kBlue);
t8->Draw();
TText *t9 = new TText(0.65, -450, "space charges alone");
t9->SetTextFont(132); t9->SetTextAngle(12.5); t9->SetTextColor(kGreen);
t9->Draw();
TLatex *t10 = new TLatex(0.15, -2800,
Form("Impurity: %.0e/cm^{3}",impurity[6]/cm3));
t10->SetTextFont(132); t10->Draw();
}
c->Print("undepleted.png");
}