5.2 Five Species Phosphorus Pair-Diffusion Model

Richardson, Carey, and Mulvaney formulated the five species phosphorus diffusion model[46,33] described by the differential equation system in (5.4) - (5.8). In these equations $P$, $I$, $V$, $E$, and $F$ represent the concentrations of substitutional phosphorus, the silicon interstitials, vacancies, phosphorus-vacancy, and phosphorus-interstitial pairs, respectively.

$$\frac{\partial V}{\partial t} = D_{V}\: \nabla ^{2}V-k^{E}_{for}\: P\: V+k^{E}_{rev}\: E-k_{bi}\: \left( V\: I-V^{eq}\: I^{eq}\right)$$ (5.4)

$$\frac{\partial I}{\partial t} = D_{I}\: \nabla ^{2}I-k^{F}_{for}\: P\: I+k^{F}_{rev}\: F-k_{bi}\: \left( V\: I-V^{eq}\: I^{eq}\right)$$ (5.5)

$$\frac{\partial E}{\partial t} = D_{E}\: \nabla ^{2}E+k^{E}_{for}\: P\: V-k^{E}_{rev}\: E$$ (5.6)

$$\frac{\partial F}{\partial t} = D_{F}\: \nabla ^{2}F+k^{F}_{for}\: P\: I-k^{E}_{rev}\: F$$ (5.7)

$$\frac{\partial P}{\partial t} = -k^{E}_{for}\: P\: V+k^{E}_{rev}\: E-k^{F}_{for}\: P\: I+k^{F}_{rev}\: F$$ (5.8)

The according MDL input deck can be structured into

• the definition of global Parameters containing the values of the coefficients
  [4] $k^{E}_{for}, k^{F}_{for}, \dots$ (Example 5.2),
• the global simulation setup for the two affected material segments (Example 5.3). To allow for series of subsequent simulation experiments the names of input and output device descriptions and the simulation times are forwarded to the simulator by using environment variables.
• the definition of the required quantities $P, V, I, E,$ and $F$ (Example 5.4). Since the electrostatic potential $\psi$ is not taken into consideration within (5.4) - (5.8) the definition of charge states can be omitted.
• The diffusion modeling setup depicted in Example 5.5
• the setup of initialization and post processing models for all affected quantities (Example 5.6 and Example 5.8), responsible for the initialization of the quantities $I$ and $V$ to their initial values of $1.0e14 cm^{-3}$.
• and finally the definition of the required coefficients of the pair diffusion model (Example 5.9 and Example 5.10).

// Parameters
Parameter<double> k_for_e = 1.0e-14;  // [cm^3/s]
Parameter<double> k_for_f = 1.0e-14;
Parameter<double> k_bi    = 1.0e-10;

Parameter<double> k_rev_e = 10.;      // [1/s]
Parameter<double> k_rev_f = 12.;

Parameter<double> Dv = - 1.0e-10;     // [cm^2/s]
Parameter<double> Di = - 1.0e-09;
Parameter<double> De = - 1.0e-13;
Parameter<double> Df = - 2.0e-13;

Parameter<double> Veq  = 1.0e14;      // [1/cm^3]
Parameter<double> Ieq  = 1.0e14;
Parameter<double> VIeq = 1.0e14 * 1.0e14;

Example 5.2: Global parameter declarations

Instance PromisNTSetup = PairDiffInitModel;
NewModel PairDiffInitModel : PromisNTSetupModel {
   evaluate {
      :inputPBF  = "$COMPILE";
      :outputPBF = "$OUTPUT";

      :endTime   = "$ENDTIME";
      :stepTime  = "$STEPTIME";

      :quantityList          = "P_Pair_Quantities";
      :chargeStateTable      = "P_Pair_Charge_States";
      :processTemperature    = "PairDiffTemperatureModel";
      :deviceSetup           = "P_Pair_Diff_DeviceSetup";

      :diffusionModel["Si"]      = "Promis";
      :diffusionModel["Ambient"] = "Promis";
      :boundaryModel["Ambient"]["Si"] = "Promis";
   }
}

NewModel PairDiffTemperatureModel : ProcessTemperatureModel {
   evaluate {
      :temp = 1173.0;
   }
}

Example 5.3: Simulator setup for the five species pair diffusion model

NewModel P_Pair_Quantities : QuantityListModel {
   evaluate {
      :quantity["P"] = {{ P }};
      :quantity["V"] = {{ V }};
      :quantity["I"] = {{ I }};
      :quantity["E"] = {{ E }};
      :quantity["F"] = {{ F }};
   }
}

NewModel P_Pair_Charge_States : ChargeStateTableModel {
   evaluate {}
}

Example 5.4: Quantity setup for the five species pair diffusion model

NewModel P_Pair_Diff_DeviceSetup : DeviceSetupModel {
   evaluate {
      :segmentPreProcessingSetup ["Si"]      = "SiSegmentPreProcessingSetup";
      :segmentCoefficientSetup   ["Si"]      = "SiCoefficient";
      :segmentPostProcessingSetup["Si"]      = "SiSegmentPostProcessingSetup";
      :segmentPreProcessingSetup ["Ambient"] = "AmbientPreprocessingSetup";
      :segmentCoefficientSetup   ["Ambient"] = "AmbientCoefficient";
      :segmentPostProcessingSetup["Ambient"] = "AmbientPostprocessingSetup";

      :boundaryCoefficientSetup["Ambient"]["Si"] = "P_Pair_BoundarySetup";
   }
}

Example 5.5: Device setup for the five species pair diffusion model

NewModel SiSegmentPreProcessingSetup : SegmentPreProcessingSetupModel {
   basicSetup {
      :quantityInitialization["P"] = "Zero_Init";
      :quantityInitialization["V"] = "Zero_Init";
      :quantityInitialization["I"] = "Zero_Init";
      :quantityInitialization["E"] = "Zero_Init";
      :quantityInitialization["F"] = "Zero_Init";
   }
   evaluate {
      call basicSetup;
      :quantityInitialization["V"] = "VI_Init";
      :quantityInitialization["I"] = "VI_Init";
   }
}

NewModel AmbientPreprocessingSetup  : SegmentPreProcessingSetupModel {
   evaluate { call basicSetup; }
}

NewModel SiSegmentPostProcessingSetup : SegmentPostProcessingSetupModel {
   setup {
      :quantityPostprocessing["P"] = "No_Post";
      :quantityPostprocessing["V"] = "No_Post";
      :quantityPostprocessing["I"] = "No_Post";
      :quantityPostprocessing["E"] = "No_Post";
      :quantityPostprocessing["F"] = "No_Post";
   }
   evaluate { call setup; }
}

NewModel AmbientPostprocessingSetup : SiSegmentPostProcessingSetup {
   evaluate { call setup; }
}

Example 5.6: Pre- and postprocessing setup

NewModel SiCoefficient : CoefficientSetupModel {
   evaluate {
      :alpha["P"]["P"] = "Uno_alpha"; :alpha["V"]["V"] = "Uno_alpha";
      :alpha["I"]["I"] = "Uno_alpha"; :alpha["E"]["E"] = "Uno_alpha";
      :alpha["F"]["F"] = "Uno_alpha";

      :a["V"]["V"] = "V_a"; :a["I"]["I"] = "I_a";
      :a["E"]["E"] = "E_a"; :a["F"]["F"] = "F_a";

      :gamma["P"] = "P_gamma";  :gamma["V"] = "V_gamma";
      :gamma["I"] = "I_gamma";  :gamma["E"] = "E_gamma";
      :gamma["F"] = "F_gamma";
   }
}

NewModel AmbientCoefficient : CoefficientSetupModel {
   evaluate {
      :alpha["P"]["P"] = "Uno_alpha"; :alpha["V"]["V"] = "Uno_alpha";
      :alpha["I"]["I"] = "Uno_alpha"; :alpha["E"]["E"] = "Uno_alpha";
      :alpha["F"]["F"] = "Uno_alpha";
   }
}


NewModel P_Pair_BoundarySetup : BoundaryCoefficientSetupModel {
   evaluate {
      :dirichlet2["V"] = 1.0e14;      :dirichlet2["I"] = 1.0e14;
      :dirichlet2["P"] = 5.0e20;      :dirichlet2["E"] = 5.0e19;
      :dirichlet2["F"] = 4.16666e19;
   }
}

Example 5.7: Coefficient setup

NewModel Zero_Init : QuantityInitializationModel {
   evaluate {
      :Cown = 0.0;
   }
}

NewModel VI_Init : QuantityInitializationModel {
   evaluate {
      :Cown = 1.0e14;
   }
}

NewModel No_Post : QuantityPostprocessingModel {}

Example 5.8: Quantity initialization and postprocessing

NewModel Uno_alpha : CoeffModel_alpha {
   evaluate { :Coeff = 1.; }
}

// Coefficients for P related equation
NewModel P_gamma : CoeffModel_gamma {
   evaluate {
     :Coeff      = - ::k_for_e * :P * :V + ::k_rev_e * :E -
                     ::k_for_f * :P * :I + ::k_rev_f * :F;
     :dCoeff d_P = - ::k_for_e * :V - ::k_for_f * :I;
     :dCoeff d_V = - ::k_for_e * :P;
     :dCoeff d_I = - ::k_for_f * :P;
     :dCoeff d_E =   ::k_rev_e;
     :dCoeff d_F =   ::k_rev_f;
   }
}


// Coefficients for V related equation
NewModel V_a : CoeffModel_a {
   evaluate { :Coeff = ::Dv; }
}

NewModel V_gamma : CoeffModel_gamma {
   evaluate {
     :Coeff      = - ::k_for_e * :P * :V + ::k_rev_e * :E -
                     ::k_bi * ( :V * :I - ::VIeq );
     :dCoeff d_P = - ::k_for_e * :V;
     :dCoeff d_V = - ::k_for_e * :P - ::k_bi * :I;
     :dCoeff d_I = - ::k_bi    * :V;
     :dCoeff d_E =   ::k_rev_e;
   }
}

// Coefficients for I related equation
NewModel I_a : CoeffModel_a {
   evaluate { :Coeff = ::Di; }
}

NewModel I_gamma : CoeffModel_gamma {
   evaluate {
     :Coeff      = - ::k_for_f * :P * :I + ::k_rev_f * :F -
                     ::k_bi * ( :V * :I - ::VIeq );
     :dCoeff d_P = - ::k_for_f * :I;
     :dCoeff d_V = - ::k_bi    * :I;
     :dCoeff d_I = - ::k_for_f * :P - ::k_bi * :V;
     :dCoeff d_F = + ::k_rev_f;
   }
}

Example 5.9: Coefficient Models (part 1)

// Coefficients for E related equation
NewModel E_a : CoeffModel_a {
   evaluate { :Coeff = ::De; }
}

NewModel E_gamma : CoeffModel_gamma {
   evaluate {     
     :Coeff      =   ::k_for_e * :P * :V - ::k_rev_e * :E;
     :dCoeff d_P =   ::k_for_e * :V;
     :dCoeff d_V =   ::k_for_e * :P;
     :dCoeff d_E = - ::k_rev_e;
   }
}

// Coefficients for F related equation
NewModel F_a : CoeffModel_a {
   evaluate { :Coeff      = ::Df;  }
}

NewModel F_gamma : CoeffModel_gamma {
   evaluate {
     :Coeff      =   ::k_for_f * :P * :I - ::k_rev_f * :F;
     :dCoeff d_P =   ::k_for_f * :I;
     :dCoeff d_I =   ::k_for_f * :P;
     :dCoeff d_F = - ::k_for_f;
   }
}

Example 5.10: Coefficient Models (part 2)

The simulation results depicted in Fig. 5.3, Fig. 5.4, and Fig. 5.5 show the expected formation of the flat point defect profiles in the inner device regions determined by their reaction term $\left( V\: I-V^{eq}\: I^{eq}\right)$ in the according equations (5.4) and (5.5). The formation of the sharp gradients of the defect concentrations near the surface induced by the Dirichlet boundary conditions and the coupling with the equations for the phosphorus, phosphorus-interstitial, and phosphorus-vacancy pair profiles in term is responsible for the creation of the phosphorus plateau, kink, and tail. The resulting profiles perfectly coincide with the results obtained by the authors of this pair diffusion model.

Figure 5.3: Dopant distribution after 6 seconds

Figure 5.4: Dopant distribution after 60 seconds

Figure 5.5: Dopant distribution after 600 seconds