Subsections

• 3.1.6.1 Artificial Boundaries
• 3.1.6.2 Semiconductor-Metal Boundaries
  • 3.1.6.2.1 Ohmic Contact
  • 3.1.6.2.2 Schottky Contact
  • 3.1.6.2.3 Polysilicon Contact
• 3.1.6.3 Insulator-Metal Boundaries
• 3.1.6.4 Semiconductor-Insulator Interface
• 3.1.6.5 Insulator-Insulator Interface
• 3.1.6.6 Semiconductor-Semiconductor Interface
  • 3.1.6.6.1 Continuous Quasi-Fermi Level Model
  • 3.1.6.6.2 Thermionic Field Emission Model
  • 3.1.6.6.3 Semiconductor-Semiconductor Thermal Interface

3.1.6 Boundary Conditions

The basic semiconductor equations are posed in a bounded domain. At the boundaries of this domain appropriate boundary conditions need to be specified for the unknowns $\psi$, $n$, $p$, $T_n$, $T_p$, and $T_{{\mathrm{L}}}$.

3.1.6.1 Artificial Boundaries

In order to separate the simulated device from neighboring devices, artificial boundaries must be specified which are not boundaries in a physical sense. The Neumann boundary condition guarantees that the simulation domain is self-contained and there are no fluxes across the boundary.

$$\mathbf{n}\cdot\mathbf{E}=0,\hspace{5mm} \mathbf{n}\cdot\mathbf{J... ...n}\cdot\mathbf{S}_{n,p}=0,\hspace{5mm} \mathbf{n}\cdot\mathbf{S}_{\mathrm{L}}=0$$ (3.20)

$$\mathbf{n}\cdot\mathbf{E}=0,\hspace{5mm} \mathbf{n}\cdot\mathbf{S}_{\mathrm{L}}=0$$ (3.21)

Here, $\mathbf{n}$ denotes an outward oriented vector normal to the boundary. (3.20) and (3.21) give the boundary conditions at the artificial boundaries for semiconductor and insulator segments, respectively.

3.1.6.2 Semiconductor-Metal Boundaries

3.1.6.2.1 Ohmic Contact

At Ohmic contacts simple Dirichlet boundary conditions apply. The contact potential $\varphi_{\mathrm{s}}$, the carrier contact concentrations $n_s$ and $p_s$, and in the HD simulation case, the contact carrier temperatures $T_n$ and $T_p$ are fixed. The metal quasi-Fermi level (which is specified by the contact voltage $\varphi_{\mathrm{m}}$) is equal to the semiconductor quasi-Fermi level. The contact potential at the semiconductor boundary reads

$$\varphi_{\mathrm{s}}= \varphi_{\mathrm{m}}+ \psi_{\mathrm{bi}}.$$ (3.22)

The built-in potential $\psi_{\mathrm{bi}}$ is calculated after [76]

$$\psi_{\mathrm{bi}} = \frac{\mathrm{k_B}\cdot T_{{\mathrm{L}}}}{\mathrm{q}} \cdot \ln \... ...\frac{1}{2\cdot C_1}\cdot \left(C+ \sqrt{C^2+4\cdot C_1\cdot C_2}\right)\right)$$

$$= -\frac{\mathrm{k_B}\cdot T_{{\mathrm{L}}}}{\mathrm{q}} \cdot \ln ... ...rac{1}{2\cdot C_2}\cdot \left(-C+ \sqrt{C^2+4\cdot C_1\cdot C_2}\right)\right),$$ (3.23)

Here, $C$ is the net concentration of dopants and other charged defects at the contact boundary. The auxiliary variables $C_1$ and $C_2$ are defined by

$$C_{1}=N_{C}\cdot \exp\left(\frac{-E_{C}}{\mathrm{k_B}\cdot T_{{\m... ...2}=N_{V}\cdot \exp\left(\frac{E_{V}}{\mathrm{k_B}\cdot T_{{\mathrm{L}}}}\right)$$ (3.24)

The carrier concentrations in the semiconductor are pinned to the equilibrium carrier concentrations at the contact. They are expressed as

$$n_s=N_{C}\cdot \exp \left(\frac{-E_{C}+q\cdot \psi_{\mathrm{bi}}}{\mathrm{k_B}\cdot T_{{\mathrm{L}}}}\right)$$ (3.25)

$$p_s=N_{V}\cdot \exp \left(\frac{E_{V}-\mathrm{q}\cdot \psi_{\mathrm{bi}}}{\mathrm{k_B}\cdot T_{{\mathrm{L}}}}\right)$$ (3.26)

The carrier temperatures $T_n$ and $T_p$ are set equal to the lattice temperature $T_{{\mathrm{L}}}$.

$$T_n = T_{{\mathrm{L}}},\hspace{5mm} T_p = T_{{\mathrm{L}}}$$ (3.27)

In the case of a thermal contact the lattice temperature $T_{{\mathrm{L}}}$ is calculated using a specified contact temperature $T_{\mathrm{C}}$ and thermal resistance $R_{\mathrm{T}}$. The thermal heat flow density $\mathbf{S}_{\mathrm{L}}$ at the contact boundary reads:

$$\mathbf{n}\cdot\mathbf{S}_{\mathrm{L}}= \frac{T_{{\mathrm{L}}}-T_{\mathrm{C}}}{R_{\mathrm{T}}}$$ (3.28)

In case no thermal resistance is specified an isothermal boundary condition is assumed and the lattice temperature $T_{{\mathrm{L}}}$ is set equal to the contact temperature $T_{\mathrm{C}}$.

$$T_{{\mathrm{L}}}= T_{\mathrm{C}}$$ (3.29)

In the case of DD simulation with self-heating an additional thermal energy is accounted for. This thermal energy is produced when the carriers have to surmount the potential difference between the conduction or valence band and the metal quasi-Fermi level. The energy equation reads:

$$\mathbf{J}_n \cdot \left(\frac{E_{C}}{\mathrm{q}}+ \varphi_{... ...}}\right) = \mathrm{div}_\mathrm {A} \mathbf{S}_{\mathrm{L}}$$ (3.30)

The expression $\mathrm{div}_\mathrm {A} \left(\mathbf{S}_{{\mathrm{L}}}\right)$ denotes the surface divergence of the thermal heat flux at the considered boundary. In the case of HD simulation with self-heating the thermal heat flow across the boundary is accounted for self-consistently.

3.1.6.2.2 Schottky Contact

At the Schottky contact mixed boundary conditions apply. The contact potential $\varphi_{\mathrm{s}}$, the carrier contact concentrations $n_s$ and $p_s$, and in the HD simulation case, the contact carrier temperatures $T_n$ and $T_p$ are fixed. The semiconductor contact potential is the difference of the metal quasi-Fermi level (which is specified by the contact voltage $\varphi_{\mathrm{m}}$) and the metal workfunction difference potential $\varphi_{\mathrm{w}}$.

$$\varphi_{\mathrm{s}}= \varphi_{\mathrm{m}}-\varphi_{\mathrm{... ...phi_{\mathrm{w}}= -\frac{\mathrm {E}_{\mathrm{w}}}{\mathrm{q}}$$ (3.31)

The difference between the conduction band energy $E_{C}$ and the metal workfunction energy gives the workfunction difference energy $\mathrm {E}_{\mathrm{w}}$ which is the so-called barrier height of the Schottky contact. The applied boundary conditions are

$$\mathbf{n}\cdot\mathbf{J}_n = -\mathrm{q}\cdot v_n \cdot (n-... ...dot\mathbf{J}_p = \mathrm{q}\cdot v_p \cdot (p-p_\mathrm {s})$$ (3.32)

Here $v_n$ and $v_p$ are the thermionic recombination velocities. The carrier concentrations are expressed as

$$n_\mathrm {s}=N_{C}\cdot\exp\left(\frac{-E_{C}-\mathrm {E}_{... ...m {E}_{\mathrm{w}}}{\mathrm{k_B}\cdot T_{{\mathrm{L}}}}\right)$$ (3.33)

The default values for $v_n$ and $v_p$ are set to 0 (see Table 3.1) which suppresses current flow through the Schottky contact ( $\mathbf{n}\cdot\mathbf{J}_{n,p}=0$).

Table 3.1: Parameter values for Schottky contact model

Material	$\mathrm {E}_{\mathrm{w}}$ [eV]	$v_n$ [m/s]	$v_p$ [m/s]
n-Si/Au	-0.55	0.0	0.0
p-Si/Au	0.55	0.0	0.0
others	0.0	0.0	0.0

The Schottky contact boundary conditions for the carrier temperatures $T_n$ and $T_p$ and the lattice temperature $T_{{\mathrm{L}}}$ are similar to the ones which apply for the Ohmic contact, i.e. (3.27) and (3.28), or respectively (3.29).

3.1.6.2.3 Polysilicon Contact

In MINIMOS-NT the polysilicon contact model after [77] is implemented. The dielectric flux $\mathrm {D}$ through the oxide reads

$$\mathbf{n}\cdot\mathbf{D} = - \frac{\varepsilon_{\mathrm{ox}}}{d_\mathrm {ox}}\cdot V_\mathrm {ox}$$ (3.34)

where $V_\mathrm {ox}$ is the voltage drop over the thin oxide layer which is introduced between polysilicon and silicon, $\varepsilon_{\mathrm{ox}}$ and $d_\mathrm {ox}$ denote respectively the permittivity and thickness of this layer. The electron and hole current densities across the contact interface read

$$\mathbf{n}\cdot\mathbf{J}_n = \sigma_\mathrm {ox} \cdot V_{ox}\\ \mathbf{n}\cdot\mathbf{J}_p = \mathrm{q}\cdot p \cdot S_p$$ (3.35)

where $\sigma_\mathrm {ox}$ is the oxide conductivity, $p$ is the hole concentration in the semiconductor, and $S_p$ is the hole surface recombination velocity. $V_\mathrm {ox}$ depends on the quasi-Fermi level in the metal (which is specified by the contact voltage $\varphi_{\mathrm{m}}$), the potential in the semiconductor $\varphi_{\mathrm{s}}$, and the built-in potential $\psi_{\mathrm{bi}}$.

$$V_\mathrm {ox} = \varphi_{\mathrm{s}}- \varphi_{\mathrm{m}}- \psi_{\mathrm{bi}}.$$ (3.36)

The polysilicon contact boundary conditions for the carrier temperatures $T_n$ and $T_p$ and the lattice temperature $T_{{\mathrm{L}}}$ are similar to the ones which apply for the Ohmic contact, i.e. (3.27) and (3.28), or respectively (3.29).

3.1.6.3 Insulator-Metal Boundaries

In the case of insulator-metal boundaries a model similar to the Schottky contact model is used to calculate the insulator contact potential. The semiconductor contact potential is the difference of the metal quasi-Fermi level (which is specified by the contact voltage $\varphi_{\mathrm{m}}$) and the metal workfunction difference potential $\Phi_\mathrm {ms}$.

$$\varphi_{\mathrm{ins}}= \varphi_{\mathrm{m}}-\Phi_\mathrm {ms}... ...hi_\mathrm {ms} = -\frac{\mathrm {E}_{\mathrm{w}}}{\mathrm{q}}$$ (3.37)

Again, $\mathrm {E}_{\mathrm{w}}$ is the workfunction difference energy. The lattice temperature is set equal to the contact temperature (3.29).

3.1.6.4 Semiconductor-Insulator Interface

In the absence of surface charges at the semiconductor-insulator interface the normal component of the dielectric displacement is continuous, and so is the potential.

$$\mathbf{n}\cdot\varepsilon_{\mathrm{s}}\cdot\mathbf{E}_\math... ... = 0 ,\hspace{5mm}\varphi_{\mathrm{s}}= \varphi_{\mathrm{ins}}$$ (3.38)

In the presence of surface charges along the interface the dielectric displacement obeys the law of Gauß

$$\mathbf{n}\cdot\varepsilon_{\mathrm{s}}\cdot\mathbf{E}_\math... ...thrm{ins}}\cdot\mathbf{E}_\mathrm {ins} = \sigma_\mathrm {s} .$$ (3.39)

At the semiconductor-insulator interface the carrier current densities (or driving forces) and the carrier heat fluxes normal to the interface vanish.

$$\mathbf{n}\cdot\mathbf{J}_{n,p}=0,\hspace{5mm}\mathrm {i.e.}\hspace{5mm}\mathbf{n}\cdot\mathbf{F}_{n,p}=0$$ (3.40)

$$\mathbf{n}\cdot\mathbf{S}_{n,p}=0$$ (3.41)

The lattice temperature at the interface is continuous.

3.1.6.5 Insulator-Insulator Interface

Similarly to the semiconductor-insulator interface (3.38) or (3.39) apply depending on the presence of surface charges. The lattice temperature must be continuous.

3.1.6.6 Semiconductor-Semiconductor Interface

The calculation of the electrostatic potential at interfaces between two semiconductor segments is similar to the one at semiconductor-insulator interfaces.

$$\varphi_{\mathrm{s1}} = \varphi_{\mathrm{s2}}$$ (3.42)

$$\mathbf{n}\cdot\varepsilon_{\mathrm{s1}}\cdot\mathbf{E}_\mathrm {... ...dot\varepsilon_{\mathrm{s2}}\cdot\mathbf{E}_\mathrm {s2} = \sigma_\mathrm {s} .$$ (3.43)

Here $\sigma_\mathrm {s}$ is the interface charge density which can be zero or non-zero. The subscripts are used to distinguish between the two semiconductor segments on both sides of the interface.

To calculate the carrier concentrations and the carrier temperatures at the interface of two semiconductor segments three different models are considered These are a model with continuous quasi-Fermi level across the interface (CQFL), a thermionic emission model (TE), and a thermionic field emission model (TFE). The derivation of these models is given in [78]. Each model can be specified separately for electrons and holes for each semiconductor-semiconductor interface.

In the following $J$ denotes the current density, $S$ the energy flux density, and $\Delta E_{\nu}$ the difference in the conduction or valence band edges, respectively. The carrier concentration is denoted by $\nu$. The subscripts denote the semiconductor segment and the carrier type.

3.1.6.6.1 Continuous Quasi-Fermi Level Model

$$\nu_{2} = \nu_{1}\cdot\left(\frac{m_{\nu 2}}{m_{\nu 1}}\right)^{3/2}\cdot \exp\left(-\frac{\Delta E_{\nu}}{\mathrm{k_B}\cdot T_{\nu 1}}\right)$$ (3.44)

$$T_{\nu 2} = T_{\nu 1}$$ (3.45)

3.1.6.6.2 Thermionic Field Emission Model

$$J_{\nu 2} = J_{\nu 1}$$ (3.46)

$$J_{\nu 2} = \mathrm{q}\cdot\left(v_{\nu 2}\cdot \nu_{2} - \frac{m_{\nu 2}}{m_... ...rac{\Delta E_{\nu} - \delta E_{\nu}}{\mathrm{k_B}\cdot T_{\nu 1}}\right)\right)$$ (3.47)

$$S_{\nu 2} = S_{\nu 1} - \frac{1}{\mathrm{q}}\cdot\left(\Delta E_{\nu} - \delta E_{\nu}\right)\cdot J_{\nu 2}$$ (3.48)

$$S_{\nu 2} = -\left(\mathrm{k_B}\cdot T_{\nu 2}\cdot v_{\nu 2}\cdot \nu_{2} - ... ...ac{\Delta E_{\nu} - \delta E_{\nu}}{\mathrm{k_B}\cdot T_{\nu1 }} \right)\right)$$ (3.49)

with the thermionic emission velocity (3.50) and the barrier height lowering (3.51).

$$v_{\nu i} = \sqrt{\frac{2\cdot\mathrm{k_B}\cdot T_{\nu i}}{\pi\cdot m_{\nu i}}}$$ (3.50)

$$\delta E_{\nu} = \left\{\begin{array}{ll} \mathrm{q}\cdot E_{\perp 2}\cdot d_\mathrm {tun}, & E_{\perp 2} > 0\\ 0,& E_{\perp 2} \leq 0 \end{array}\right.$$ (3.51)

The barrier height lowering depends on the electric field orthogonal to the interface $E_{\perp 2}$ and the effective tunneling length $d_\mathrm {tun}$. For $\delta E_{\nu} = 0$ the TFE model reduces to the TE model.

By using the CQFL model a Dirichlet interface condition is applied. The carrier concentrations are directly determined in a way that the quasi-Fermi level across the interface remains continuous. The model is suitable for use at homojunctions. However, it is erroneous to assume continuous quasi-Fermi levels at abrupt heterojunctions. Also the bandgap alignment of the adjustent semiconductors is ignored when such continuous condition is enforced. Therefore, models using a Neumann interface condition, like the TFE model or the TE model, which determine the current flux across the interface, must be used. Modeling the electron and hole current as well as the energy flux across heterointerfaces is a complex task. Several models for different types of interfaces have been proposed [79,80,81,82]. The TE model is commonly used to model the current across heterojunctions of compound semiconductors. The TFE model extends the TE model by accounting for tunneling effects through the heterojunction barrier by introducing a field dependent barrier height lowering. In [83] a method for unified treatment of interface models was presented. It allows a change of the interface condition from Neumann to Dirichlet type in the limit case of very strong barrier reduction due to tunneling.

3.1.6.6.3 Semiconductor-Semiconductor Thermal Interface

The lattice temperature is assumed to be continuous across semiconductor-semiconductor interfaces. In the case of DD simulation with self-heating an additional thermal energy is accounted for at heterojunction interfaces. This thermal energy is produced when the carriers have to surmount the energy difference in the conduction and valence bands, $\Delta E_{C}$ and $\Delta E_{V}$, respectively. The energy equation reads:

$$\frac{\mathbf{J}_n}{\mathrm{q}} \cdot \Delta E_{C}+ \frac{\m... ... = {\mathop{\rm div_A}}\left(\mathbf{S}_{{\mathrm{L}}}\right)$$ (3.52)

The expression $\mathrm{div}_\mathrm {A} \left(\mathbf{S}_{{\mathrm{L}}}\right)$ denotes the surface divergence of the thermal heat flux at the considered boundary. In the case of HD simulation with self-heating the thermal heat flow across the boundary is accounted for self-consistently.