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
// https://link.springer.com/chapter/10.1007/10832182_519
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");
   gStyle->SetPadRightMargin(0.01);
   gStyle->SetPadLeftMargin(0.11);
   gStyle->SetPadTopMargin(0.01);
   gStyle->SetPadBottomMargin(0.12);

   // 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();
   gPad->SetLogx(); gPad->SetGridx(); gPad->SetGridy();
   gPad->Print("depleted.png");

   // 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");
}